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PROFESSOR: You'll recall, at
least some of you will recall,

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that last time we ended up
looking at a simulation of a

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drunken university student
wandering randomly in a field.

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I'm going to return to that.


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Before I get to the interesting
part, I do want to call your

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attention to something that
some people seemed a little

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00:00:45 --> 00:00:47
bit confused by last time.


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So you remember that we
had this thing called

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perform trial.


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And the way we basically tested
the drunk, what would happen,

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is we would call this, passing
in a time, the number of steps,

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the amount of time that the
drunk was wondering,

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and the field.


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And there was just 1 line,
that I blazed past,

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probably too quickly.


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So what that line is saying, is
we, from the field, we're going

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to call the method get
drunk, which is going

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to return us a drunk.


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And then we're going to
call the move method for

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that drunk, passing the
field as an argument.

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Now the question is, what would
have happened if I had left

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off these parentheses?


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There.


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A volunteer?


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What would, we will do the
experiment in a minute.

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And by the way, I want to
emphasize the importance

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of experimentation.


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I was kind of surprised, for
example, to get email from

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people saying, well, what
was the correct answer to

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00:02:09 --> 00:02:12
question 3 on the quiz?


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And my sort of response
was, why don't you

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00:02:14 --> 00:02:18
type it and find out?


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When in doubt, run
the experiment.

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So, what's going to happen,
what would have happened,

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had I failed to type that
pair of parentheses?

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Anybody?


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00:02:34 --> 00:02:43
Well, what is, get drunk?


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It is a method.


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And remember, in Python,
methods, like classes,

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are themselves objects.


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So get drunk would have a
returned an object which

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was a method, and then,
well, let's try it.

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And the error message will tell
us everything we need to know.

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Function has no attribute move.


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Sure enough, the method does
not have an attribute move.

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The instance of the class
has the attribute.

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And so what those parentheses
did, was tell me to invoke the

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method get drunk, so now
instead of that being the

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method itself, it's the result
returned by the method.

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Which is an instance of class
drunk, and that instance has

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an attribute move, which
is itself a method.

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This is a very common
programming paradigm, and it's

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important to sort of lock into
your heads, the distinction

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between the method and an
invocation of the method.

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Because you can write either,
and sometimes you won't get an

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error message, you'll just
get the wrong answer.

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And that's kind of bad.


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That make sense to everybody?


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Does it not make sense to
anybody, maybe is the

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question I should ask?


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All right, we'll go on, and
we'll see more examples

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of this kind of thing.


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Now last week, I ran this
several times to see what would

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happen, and we saw that in fact
contrary to everyone's, most

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everyone's, expectation, the
longer we ran the simulation,

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the further the drunk was
from the starting point.

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And we saw that by plotting
how far the drunk was

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after each time unit.


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We ran it several times,
we got different answers.

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And at that point we should ask
ourselves, is that really the

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way to go about answering
the original question?

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Which was, how far should
we expect the drunk to be?

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The answer is no.


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I don't want to sit there at
the keyboard typing in 400

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examples and then, in my head,
trying to figure out what's

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quote, typical, unquote.


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Instead, I need to organize my
simulation so that it runs a

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number of trials for me and
summarizes the results.

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All the simulations we're going
to look at it, in fact almost

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all the simulations anyone will
ever write, sort of, have

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the same kind of structure.


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We start with an inner loop
that simulates one trial.

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That's what we have here,
right, we have happen

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to have a function.


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So when I say inner loop, what
I mean is, not that I'll write

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a program a bunch of nested
loops, but that I'll write a

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program with some function
calls, and down at sort of

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the bottom of the stack
will be perform trial.

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Which stimulates one
trial of some number of

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seconds, in this case.


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And then I'll quote, enclose,
unquote the inner loop in

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00:07:04 --> 00:07:25
another loop that conducts an
appropriate number of trials.

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00:07:25 --> 00:07:29
Now a little bit later in the
term, we'll get to the question

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of, how do we know what
an appropriate number is?

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00:07:33 --> 00:07:39
For today, we'll
just say, a lot.

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00:07:39 --> 00:07:42
And we'll talk about
that a little bit more.

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00:07:42 --> 00:08:00
And then finally, what we want
to do, is calculate and present

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00:08:00 --> 00:08:07
some relevant statistics
about the trials.

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00:08:07 --> 00:08:11
And we'll talk about
what's relevant.

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00:08:11 --> 00:08:17
I want to emphasize today
the presentation of them.

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00:08:17 --> 00:08:21
Last time we looked at a graph,
which I think you'll all agree,

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00:08:21 --> 00:08:33
was a lot prettier to look at
an array of, say, 1000 numbers.

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00:08:33 --> 00:08:34
All right.


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00:08:34 --> 00:08:41
So now, on your handout and
on the screen, you'll see

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00:08:41 --> 00:08:43
the way we've done this.


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00:08:43 --> 00:08:46
So perform trial, we've already
seen, so we know what the

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00:08:46 --> 00:08:49
inner loop looks like.


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00:08:49 --> 00:08:53
We can ignore first test, that
was just my taking the code

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00:08:53 --> 00:08:58
that I had in line last time
and putting it in a function.

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00:08:58 --> 00:09:02
And now what you'll see in
the handout is perform

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00:09:02 --> 00:09:07
sim and answer question.


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00:09:07 --> 00:09:11
So, let's look it those.


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00:09:11 --> 00:09:17
So perform sim, sim for
simulation, takes the amount of

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00:09:17 --> 00:09:21
time, the number of steps for
each trial, plus the

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00:09:21 --> 00:09:26
number of trials.


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00:09:26 --> 00:09:32
It starts with an empty list,
dist short for distances here,

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00:09:32 --> 00:09:35
saying so far we haven't run
any trials, we don't know what

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any of the distances are.


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00:09:37 --> 00:09:42
And then for trial in range
number of trials, it creates a

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drunk, and I'm putting the
trial as part of the drunk's

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00:09:50 --> 00:09:53
name, just so we can make sure
that the drunks are

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00:09:53 --> 00:09:56
all different.


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00:09:56 --> 00:09:59
Then I'm going to create
a field with that drunk

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00:09:59 --> 00:10:03
in it at location 0,0.


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00:10:03 --> 00:10:08
And then I'm going to call
perform trial with the

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00:10:08 --> 00:10:14
time in that field to
get the distances.

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00:10:14 --> 00:10:22
What does perform trial return?


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00:10:22 --> 00:10:25
We looked at it earlier.


137
00:10:25 --> 00:10:30
What is perform
trial returning?

138
00:10:30 --> 00:10:35
Somebody?


139
00:10:35 --> 00:10:39
Surely someone can
figure this one out.

140
00:10:39 --> 00:10:45
I have a whole new bag of
candy here, post-Halloween.

141
00:10:45 --> 00:10:47
What kind of thing
is it returning?

142
00:10:47 --> 00:10:49
Is it returning a number?


143
00:10:49 --> 00:10:51
If you think it's returning
a number, oh, ok, you gonna

144
00:10:51 --> 00:10:55
tell me what kind of thing
it's returning, please?

145
00:10:55 --> 00:10:57
A list, thank you.


146
00:10:57 --> 00:11:02
And it's a list of what?


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00:11:02 --> 00:11:04
I'm giving you the candy
on spec, assuming you'll

148
00:11:04 --> 00:11:07
give me the right answer.


149
00:11:07 --> 00:11:10
List of distances, exactly.


150
00:11:10 --> 00:11:13
So how far way it is
after each time step?

151
00:11:13 --> 00:11:19
This is exactly the list
that we graphed last time.

152
00:11:19 --> 00:11:27
So perform sim will get that
list and append it to the list.

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00:11:27 --> 00:11:34
So dist list will be a list of
lists, each element will be

154
00:11:34 --> 00:11:38
a list of distances, right?


155
00:11:38 --> 00:11:42
OK, so that's good.


156
00:11:42 --> 00:11:50
And that's now step 2.


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00:11:50 --> 00:11:54
In some sense, all of this
is irrelevant to the

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00:11:54 --> 00:11:58
actual question we started
with, in some sense.

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00:11:58 --> 00:12:00
This is just the structure.


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00:12:00 --> 00:12:03
Then I'm going to write a
function called answer

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00:12:03 --> 00:12:08
question, or ans quest,
designed to actually address

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00:12:08 --> 00:12:12
the original question.


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00:12:12 --> 00:12:14
So it, too, is going
to create a list.

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00:12:14 --> 00:12:19
This is a list called means.


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00:12:19 --> 00:12:29
Then it's going to call perform
sim, to get this list of lists,

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00:12:29 --> 00:12:36
then it will go through that
and calculate the means and

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00:12:36 --> 00:12:40
create a list of means.


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00:12:40 --> 00:12:43
And then it will plot it, and
we'll get to, before this

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00:12:43 --> 00:12:45
lecture's over, how
the plotting works.

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00:12:45 --> 00:12:49
So I'm following exactly
that structure here.

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It calls this function, which
runs an appropriate number of

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trials by calling that
function, and then we'll

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calculate and present
some statistics.

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00:13:01 --> 00:13:11
So now let's run it.


175
00:13:11 --> 00:13:12
All right, so what have I done?


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00:13:12 --> 00:13:15
I typed an inadvertent chara --
ah, yes, I typed an s which

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I didn't intend to type.


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00:13:30 --> 00:13:33
It's going to take a little
while, it's loading Pylab.

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00:13:33 --> 00:13:37
Now it's running
the simulation.

180
00:13:37 --> 00:13:43
All right, and
here's a picture.

181
00:13:43 --> 00:13:50
So, when we ran it, let's look
at the code for a minute here,

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00:13:50 --> 00:13:55
what we can see is at the
bottom, I called ans quest,

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00:13:55 --> 00:14:04
saying each trial should be 500
steps, and run 100 trials.

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00:14:04 --> 00:14:20
And then we'll plot this graph.


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00:14:20 --> 00:14:23
Graph is lurking somewhere
in there, it is.

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00:14:23 --> 00:14:26
And one of the nice things
we'll see is, it's kind of

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00:14:26 --> 00:14:31
smooth, And we'll come back to
this, but the fact that it's

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00:14:31 --> 00:14:38
sort of smooth makes me feel
that running 100 trials might

189
00:14:38 --> 00:14:43
actually be enough to give
me a consistent answer.

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00:14:43 --> 00:14:45
You know, if it had been
bouncing up and down as

191
00:14:45 --> 00:14:51
we went, then we'd say,
jeez, no trend here.

192
00:14:51 --> 00:14:56
Seeing a relatively smooth
trend makes me feel somewhat

193
00:14:56 --> 00:15:01
comfortable that we're actually
getting an appropriate answer.

194
00:15:01 --> 00:15:04
And that if I were to run 500
trials, the line would be

195
00:15:04 --> 00:15:08
smoother, but it would
look kind of the same.

196
00:15:08 --> 00:15:12
Because it doesn't look like
it's moving here, in arbitrary

197
00:15:12 --> 00:15:14
directions large amounts.


198
00:15:14 --> 00:15:19
It's not like the stock market.


199
00:15:19 --> 00:15:23
Should I be happy?


200
00:15:23 --> 00:15:25
I've sort of done what I
wanted, I kind of I think I

201
00:15:25 --> 00:15:30
have an answer now, which is
500 steps, it should be four

202
00:15:30 --> 00:15:37
and a half units away
from the origin.

203
00:15:37 --> 00:15:39
What do you think?


204
00:15:39 --> 00:15:48
Who think this is
the right answer?

205
00:15:48 --> 00:15:55
So who thinks it's a wrong
answer, raise your hand?

206
00:15:55 --> 00:15:57
All right, TAs, what
do you guys think?

207
00:15:57 --> 00:15:59
Putting you on the spot.


208
00:15:59 --> 00:16:03
Right answer or wrong answer?


209
00:16:03 --> 00:16:06
They think it's right.


210
00:16:06 --> 00:16:11
Well, shame on them.


211
00:16:11 --> 00:16:17
Let's remember, rack our brains
to a week ago, when we ran a

212
00:16:17 --> 00:16:19
bunch of individual tests.


213
00:16:19 --> 00:16:22
And let's see what we
get if we do that.

214
00:16:22 --> 00:16:27
And the point here is, it's
always good to check.

215
00:16:27 --> 00:16:30
My recollection, when I
looked at this, was that

216
00:16:30 --> 00:16:32
something was amiss.


217
00:16:32 --> 00:16:36
Because I kind of remember,
when I ran the test last

218
00:16:36 --> 00:16:41
time, we were more like
40 away than four away.

219
00:16:41 --> 00:16:44
Well all right, let's try it.


220
00:16:44 --> 00:16:58
We'll, sometimes happens, all
right, I'm going to have to

221
00:16:58 --> 00:17:03
restart Idol here, just, as you
all, at least all who use

222
00:17:03 --> 00:17:06
Macintoshes know, this
happens sometimes,

223
00:17:06 --> 00:17:18
it's not catastrophic.


224
00:17:18 --> 00:17:22
Sigh.


225
00:17:22 --> 00:17:26
So this reminds me
of the old joke.

226
00:17:26 --> 00:17:31
That a computer scientist, a
mechanical engineer, were

227
00:17:31 --> 00:17:36
riding in a car and the car
stalled, stopped running.

228
00:17:36 --> 00:17:39
And the mechanical engineer
said I know what to do,

229
00:17:39 --> 00:17:42
let's go out and check the
carburetor, and look

230
00:17:42 --> 00:17:43
at the engine.


231
00:17:43 --> 00:17:47
The computer scientist said, no
that's the wrong thing to do.

232
00:17:47 --> 00:17:50
What you ought to do is, let's
turn off the key, get out of

233
00:17:50 --> 00:17:53
the car, shut the doors, open
the doors, get back

234
00:17:53 --> 00:17:56
in and restart it.


235
00:17:56 --> 00:17:58
And sure enough, it worked.


236
00:17:58 --> 00:18:03
So when in doubt, reboot.


237
00:18:03 --> 00:18:09
So, we'll come down, we'll do
that, and we're going to call

238
00:18:09 --> 00:18:18
first test here, and see
what that gives us.

239
00:18:18 --> 00:18:33
And we'll, for the
moment, ignore that.

240
00:18:33 --> 00:18:35
Well, look at this.


241
00:18:35 --> 00:18:38
We ran a bunch of Homer's
random walks, and maybe it

242
00:18:38 --> 00:18:45
isn't 40, but not even
one of them was four.

243
00:18:45 --> 00:18:50
So now we see is, we've run
two tests, and we've gotten

244
00:18:50 --> 00:18:54
inconsistent answers.


245
00:18:54 --> 00:18:57
Well, we don't know
which one is wrong.

246
00:18:57 --> 00:18:59
We know that one
of them is wrong.

247
00:18:59 --> 00:19:01
We don't even know that,
maybe we just got unlucky

248
00:19:01 --> 00:19:04
with these five tests.


249
00:19:04 --> 00:19:10
But odds are, something is
wrong, that there's a bug here.

250
00:19:10 --> 00:19:15
And, we have to figure
out which one.

251
00:19:15 --> 00:19:20
So how would we go
about doing that?

252
00:19:20 --> 00:19:28
Well, I'm going to do what I
always recommend people do.

253
00:19:28 --> 00:19:34
Which was, find a really simple
example, One for which I

254
00:19:34 --> 00:19:38
actually know the answer.


255
00:19:38 --> 00:19:40
So what would be a good
example for which I

256
00:19:40 --> 00:19:49
might know the answer?


257
00:19:49 --> 00:19:54
Give me the simplest example of
a simulation of the random walk

258
00:19:54 --> 00:19:57
I could run, where you're
confident you know

259
00:19:57 --> 00:19:59
what the answer is.


260
00:19:59 --> 00:20:04
Yeah? one step, exactly,
and what's the answer

261
00:20:04 --> 00:20:07
after one step?


262
00:20:07 --> 00:20:08
One.


263
00:20:08 --> 00:20:12
She can't catch and
talk at the same time.

264
00:20:12 --> 00:20:12
Exactly.


265
00:20:12 --> 00:20:17
So we know if we simulate it,
one, the drunk has moved in

266
00:20:17 --> 00:20:19
some direction, and is
going to be exactly one

267
00:20:19 --> 00:20:21
step from the origin.


268
00:20:21 --> 00:20:25
So now we can go and
see what we get.

269
00:20:25 --> 00:20:27
So let's do that.


270
00:20:27 --> 00:20:36
And we'll change this
to be one, and we'll

271
00:20:36 --> 00:20:45
change this to be one.


272
00:20:45 --> 00:20:56
We'll see what the answer is.


273
00:20:56 --> 00:20:59
Well.


274
00:20:59 --> 00:21:02
50?


275
00:21:02 --> 00:21:06
Well, kind of makes me worry.


276
00:21:06 --> 00:21:08
1.


277
00:21:08 --> 00:21:14
All right, so we see that the
simple test of Homer gives

278
00:21:14 --> 00:21:15
me the right answer.


279
00:21:15 --> 00:21:17
We don't know it's always
the right answer, we know

280
00:21:17 --> 00:21:18
it's, at least for this 1.


281
00:21:18 --> 00:21:22
But we know the other
1 was just way off.

282
00:21:22 --> 00:21:25
Interestingly, unlike the
previous time, instead of being

283
00:21:25 --> 00:21:29
way too low, it's way too high.


284
00:21:29 --> 00:21:33
So that gives us some
pause for thought.

285
00:21:33 --> 00:21:37
And now we need to
go in and debug it.

286
00:21:37 --> 00:21:40
So let's go and debug it.


287
00:21:40 --> 00:21:51
And seems to me the right
thing to do is to go here.

288
00:21:51 --> 00:21:56
Oh, boy, I'm going to have to
restart it again, not good.

289
00:21:56 --> 00:22:00
And we'll put an
intermediate value.

290
00:22:00 --> 00:22:07
Actually, maybe we'll do it.


291
00:22:07 --> 00:22:16
What would be a nice
thing to do here?

292
00:22:16 --> 00:22:18
We're going to come here.


293
00:22:18 --> 00:22:21
Well, let's, you know, we want
to go somewhere halfway through

294
00:22:21 --> 00:22:24
the program, print some
intermediate value that will

295
00:22:24 --> 00:22:27
give us some information.


296
00:22:27 --> 00:22:35
So, this might be a good place.


297
00:22:35 --> 00:22:43
And, what should we print?


298
00:22:43 --> 00:22:55
Well, what values do you
think you should get here?

299
00:22:55 --> 00:22:55
Pardon?


300
00:22:55 --> 00:22:57
STUDENT: The total
distance so far.

301
00:22:57 --> 00:23:02
PROFESSOR: The total
distance so far.

302
00:23:02 --> 00:23:07
So that would be a
good thing to print.

303
00:23:07 --> 00:23:24
And what do we think
it should be?

304
00:23:24 --> 00:23:29
We'll comment this 1 out since
we think that works, and just

305
00:23:29 --> 00:23:33
to be safe, let's not even run
100 trials let's, run one

306
00:23:33 --> 00:23:38
trial, or two trials maybe.


307
00:23:38 --> 00:23:45
See we get.


308
00:23:45 --> 00:23:49
0 and then 2.


309
00:23:49 --> 00:23:51
W, 0 was sort of what
we expected the first

310
00:23:51 --> 00:23:54
time around, but 2?


311
00:23:54 --> 00:24:00
How did you get to be 2?


312
00:24:00 --> 00:24:03
Anyone want to see
what's going on here?

313
00:24:03 --> 00:24:08
So we see, right here we
have the wrong answer.

314
00:24:08 --> 00:24:18
Well, maybe we should see what
things looked like before this?

315
00:24:18 --> 00:24:19
Is it the lists are wrong?


316
00:24:19 --> 00:24:23
What am I doing wrong here?


317
00:24:23 --> 00:24:32
I'll bet someone can
figure this out.

318
00:24:32 --> 00:24:32
Pardon?


319
00:24:32 --> 00:24:35
STUDENT: [INAUDIBLE]


320
00:24:35 --> 00:24:39
PROFESSOR: Well, I'm adding
them up, fair enough.

321
00:24:39 --> 00:24:42
But so tot looks OK.


322
00:24:42 --> 00:24:48
So, all right, maybe we
should take a look at means.

323
00:24:48 --> 00:24:49
Right?


324
00:24:49 --> 00:25:03
Let's take a look at
what that looks like.

325
00:25:03 --> 00:25:06
Not bad.


326
00:25:06 --> 00:25:08
All right, so maybe my
example's too simple.

327
00:25:08 --> 00:25:19
Let's try a little bit bigger.


328
00:25:19 --> 00:25:26
Hmmm -- 2.5?


329
00:25:26 --> 00:25:30
All right, so now I know what's
going wrong is, somehow not

330
00:25:30 --> 00:25:34
that I'm messing up tot, but
that I'm computing the

331
00:25:34 --> 00:25:37
mean incorrectly.


332
00:25:37 --> 00:25:39
Where am I computing the mean?


333
00:25:39 --> 00:25:41
They're only two
expressions here.

334
00:25:41 --> 00:25:44
There's tot, we've
checked that.

335
00:25:44 --> 00:25:48
So there must be a problem
with the divisor, that's

336
00:25:48 --> 00:25:51
the only thing that's left.


337
00:25:51 --> 00:25:54
Yeah?


338
00:25:54 --> 00:25:55
STUDENT: [INAUDIBLE]


339
00:25:55 --> 00:25:56
PROFESSOR: Exactly right.


340
00:25:56 --> 00:26:01
I should be dividing by
the length of the list.

341
00:26:01 --> 00:26:06
The number of things
I'm adding to tot.

342
00:26:06 --> 00:26:19
So I just, inadvertently,
divided by, I have

343
00:26:19 --> 00:26:21
a list of lists.


344
00:26:21 --> 00:26:23
And what I really wanted
to do is, divide by

345
00:26:23 --> 00:26:25
the number of lists.


346
00:26:25 --> 00:26:28
Because I'm computing the mean
for each list, adding it to

347
00:26:28 --> 00:26:31
total, and then at the end I
need to divide by the number of

348
00:26:31 --> 00:26:36
lists who's means I computed,
not by the length of 1

349
00:26:36 --> 00:26:41
of the lists, right?


350
00:26:41 --> 00:26:49
So now, let's see what
happens if we run it.

351
00:26:49 --> 00:26:52
Now, we get some output
printed, which I really

352
00:26:52 --> 00:26:57
didn't want, but it happens.


353
00:26:57 --> 00:27:02
Well this looks a lot better.


354
00:27:02 --> 00:27:02
Right?


355
00:27:02 --> 00:27:08
Sure enough, it's 1.


356
00:27:08 --> 00:27:10
All right, so now
I'm feeling better.

357
00:27:10 --> 00:27:13
I'm going to get rid of this
print statement, if we're gonna

358
00:27:13 --> 00:27:16
run a more extensive test.


359
00:27:16 --> 00:27:25
And now we can go back to
our original question.

360
00:27:25 --> 00:27:35
And run it.


361
00:27:35 --> 00:27:37
Well, this looks a lot
more consistent with

362
00:27:37 --> 00:27:41
what we saw before.


363
00:27:41 --> 00:27:49
It says that on average,
you should be around 20.

364
00:27:49 --> 00:27:52
So we feel pretty
good about that.

365
00:27:52 --> 00:27:55
Now, just to feel even better,
I'm going to double the

366
00:27:55 --> 00:28:10
number of trials and
see what that tells us.

367
00:28:10 --> 00:28:12
And it's still around 20.


368
00:28:12 --> 00:28:15
Line a little smoother.


369
00:28:15 --> 00:28:18
And if I where do 1000 trials
will get a little smoother, and

370
00:28:18 --> 00:28:21
it would still be around 20.


371
00:28:21 --> 00:28:26
Maybe slightly different each
time, but consistent with what

372
00:28:26 --> 00:28:29
we saw before, when we
ran the other program.

373
00:28:29 --> 00:28:34
We can feel that we're actually
doing something useful.

374
00:28:34 --> 00:28:38
And so now we can conclude, and
would actually be the correct

375
00:28:38 --> 00:28:43
conclusion, that we know about
how far this random drunk is

376
00:28:43 --> 00:28:47
going to move in 500 steps.


377
00:28:47 --> 00:28:50
And if you want to know how far
he would move in 1000 steps,

378
00:28:50 --> 00:28:54
we could try that, too.


379
00:28:54 --> 00:28:55
All right.


380
00:28:55 --> 00:28:59
What are the lessons here?


381
00:28:59 --> 00:29:05
One lesson is to look at
the labels on the axes.

382
00:29:05 --> 00:29:08
Because if we just looked at
it without noticing these

383
00:29:08 --> 00:29:13
numbers, it looks the same.


384
00:29:13 --> 00:29:13
Right?


385
00:29:13 --> 00:29:15
This doesn't look any
different, in some sense, than

386
00:29:15 --> 00:29:18
when the numbers were four.


387
00:29:18 --> 00:29:20
So you can't just look at
the shape of the curve, you

388
00:29:20 --> 00:29:22
have to look at the values.


389
00:29:22 --> 00:29:25
So what does that tell me?


390
00:29:25 --> 00:29:30
It tells me that a responsible
person will always

391
00:29:30 --> 00:29:32
label the axes.


392
00:29:32 --> 00:29:39
As I have done here, not only
giving you the numbers, but

393
00:29:39 --> 00:29:42
telling you it's the distance.


394
00:29:42 --> 00:29:46
I hate it when I look at graphs
and I have to guess what

395
00:29:46 --> 00:29:48
the x- and y- axes are.


396
00:29:48 --> 00:29:52
Here it says time versus
distance, and you also

397
00:29:52 --> 00:29:55
notice I put a title on it.


398
00:29:55 --> 00:30:00
So there's a point there.


399
00:30:00 --> 00:30:08
And look, when you're doing it.


400
00:30:08 --> 00:30:16
Ask if the answer make sense.


401
00:30:16 --> 00:30:20
One of the things we'll see as
we go on, is you can get all

402
00:30:20 --> 00:30:24
your statistics right, and
still get the wrong answer

403
00:30:24 --> 00:30:29
because of a consistent bug.


404
00:30:29 --> 00:30:33
And so always just say, do I
believe it, or is this so

405
00:30:33 --> 00:30:37
counterintuitive that
I'm suspicious?

406
00:30:37 --> 00:30:51
And as part of that ask, is it
consistent with other evidence?

407
00:30:51 --> 00:30:54
In this case we had the
evidence of watching

408
00:30:54 --> 00:30:56
an individual walk.


409
00:30:56 --> 00:30:59
Now those two things were
not consistent, don't know

410
00:30:59 --> 00:31:03
which is wrong, but it
must be one of them.

411
00:31:03 --> 00:31:07
And then the final point I
wanted to make, is that you can

412
00:31:07 --> 00:31:20
be pretty systematic about
debugging, And in particular,

413
00:31:20 --> 00:31:28
debug with a simple example.


414
00:31:28 --> 00:31:32
Right, instead of trying to
debug 500 steps and 100 trials,

415
00:31:32 --> 00:31:35
I said, all right, let's look
at one step and four

416
00:31:35 --> 00:31:38
trials, five trials.


417
00:31:38 --> 00:31:41
OK, where in my head I knew
what it should look like,

418
00:31:41 --> 00:31:44
and then I could check it.


419
00:31:44 --> 00:31:47
All right.


420
00:31:47 --> 00:31:52
Jumping up a level or
three of abstraction now.

421
00:31:52 --> 00:31:54
What we've done, is we've
introduced the notion of a

422
00:31:54 --> 00:32:00
random walk in the context of
a pretty contrived example.

423
00:32:00 --> 00:32:06
But in fact, it's worth knowing
that random walks are used all

424
00:32:06 --> 00:32:11
over the place to solve real
problems, deal with

425
00:32:11 --> 00:32:13
real phenomena.


426
00:32:13 --> 00:32:28
So for example, if you look at
something like Brownian motion,

427
00:32:28 --> 00:32:32
which can be used to model the
path traced by a molecule as it

428
00:32:32 --> 00:32:35
travels in a liquid or a gas.


429
00:32:35 --> 00:32:41
Typically, people who do that
model it using a random walk.

430
00:32:41 --> 00:32:45
And, depending upon, say the
density of the gas or the

431
00:32:45 --> 00:32:49
liquid, the size of the
molecules, they change

432
00:32:49 --> 00:32:53
parameters in the simulation,
how far it, say, goes in each

433
00:32:53 --> 00:32:55
unit time and things like that.


434
00:32:55 --> 00:32:58
But they use a random walk
to try and model what

435
00:32:58 --> 00:33:02
will really happened.


436
00:33:02 --> 00:33:06
People have attempted, for
several hundred years now, to

437
00:33:06 --> 00:33:11
use, well, maybe a 150 years,
to use random walks to

438
00:33:11 --> 00:33:12
model the stock market.


439
00:33:12 --> 00:33:17
There was a very famous book
called A Random Walk Down Wall

440
00:33:17 --> 00:33:24
Street, that argued that things
happened as a, random walk was

441
00:33:24 --> 00:33:29
a good way to model things.


442
00:33:29 --> 00:33:32
There's a lot of evidence that
says that's wrong, but people

443
00:33:32 --> 00:33:36
continue to attempt to do it.


444
00:33:36 --> 00:33:44
They use it a lot in biology to
do things like model kinetics.

445
00:33:44 --> 00:33:50
So, the kinetics of a protein,
DNA strand exchange,

446
00:33:50 --> 00:33:52
things of that nature.


447
00:33:52 --> 00:33:55
A separation of
macro-molecules, the movement

448
00:33:55 --> 00:34:02
of microorganisms all of those
things are done in biology.

449
00:34:02 --> 00:34:04
And do that.


450
00:34:04 --> 00:34:10
People use it to
model evolution.

451
00:34:10 --> 00:34:15
They look at mutations as
kind of a random event.

452
00:34:15 --> 00:34:20
So, we'll come back to this,
but random walks are used over

453
00:34:20 --> 00:34:24
and over and over again in the
sciences, the social sciences

454
00:34:24 --> 00:34:29
and therefore a very useful
thing to notice about.

455
00:34:29 --> 00:34:30
All right, we're going
to come back to that.

456
00:34:30 --> 00:34:34
We're going to even come back
to our drunken student and look

457
00:34:34 --> 00:34:37
at other kinds of random walks
other than the kind

458
00:34:37 --> 00:34:38
we just looked at.


459
00:34:38 --> 00:34:42
Before I do that, though, I
wanted back up and take the

460
00:34:42 --> 00:34:45
magic out of plotting.


461
00:34:45 --> 00:34:50
So we've gone from the sublime,
of what random walks are good

462
00:34:50 --> 00:34:54
for, to in some sense the
ridiculous, the actual

463
00:34:54 --> 00:34:56
syntax for plotting things.


464
00:34:56 --> 00:35:01
And maybe it's not
ridiculous, but it's boring.

465
00:35:01 --> 00:35:05
But you need it, so
let's look at it.

466
00:35:05 --> 00:35:12
So we're doing this using a
package called Pylab, which is

467
00:35:12 --> 00:35:28
in itself built on a package
called Pylab, either pronounced

468
00:35:28 --> 00:35:32
num p or num pi, you can choose
your pronunciation

469
00:35:32 --> 00:35:34
as you prefer.


470
00:35:34 --> 00:35:38
This basically gives you a lot
of operations on numbers,

471
00:35:38 --> 00:35:43
numbered things, and on top of
that, someone bill Pylab which

472
00:35:43 --> 00:35:47
is designed to provide a Python
interface to a lot of the

473
00:35:47 --> 00:35:50
functionality you
get in Matlab.

474
00:35:50 --> 00:35:55
And in particular, we're going
to be using today the plotting

475
00:35:55 --> 00:36:00
functionality that comes with
Matlab, or the version of it.

476
00:36:00 --> 00:36:05
So we're going to say, from
Pylab import star, that's

477
00:36:05 --> 00:36:09
just so I don't have to type
Pylab dot plot every time.

478
00:36:09 --> 00:36:12
And I'm going import random
which we're going to use later.

479
00:36:12 --> 00:36:15
So let's look at it now.


480
00:36:15 --> 00:36:19
First thing we're going to do
is plot 1, 2, 3, 4, and then

481
00:36:19 --> 00:36:24
1, 2, 3, and then 5, 6, 7, 8.


482
00:36:24 --> 00:36:28
And then at the very bottom,
you'll see this line show.

483
00:36:28 --> 00:36:31
That's going to annoy the heck
out of you throughout the

484
00:36:31 --> 00:36:34
semester, the rest
of the semester.

485
00:36:34 --> 00:36:37
Because what happens is, Pylab
produces all these beautiful

486
00:36:37 --> 00:36:40
plots, and then does not
display them until

487
00:36:40 --> 00:36:43
you type show.


488
00:36:43 --> 00:36:47
So remember, at the end of
every program, kind of,

489
00:36:47 --> 00:36:50
the last thing you should
execute should be show.

490
00:36:50 --> 00:36:54
You don't want to execute it in
the middle, because what

491
00:36:54 --> 00:36:57
happens in the middle is it, in
an interactive mode at

492
00:36:57 --> 00:37:00
least, it just stops.


493
00:37:00 --> 00:37:03
And displays the graphs, and
until you make the plots go

494
00:37:03 --> 00:37:07
away, it won't execute
the next line.

495
00:37:07 --> 00:37:09
Which is why I've tucked the
show at the very bottom

496
00:37:09 --> 00:37:11
of my script here.


497
00:37:11 --> 00:37:14
Inevitably, you will
forget to type show.

498
00:37:14 --> 00:37:17
You will ask a TA, how come my
graphs aren't appearing in the

499
00:37:17 --> 00:37:21
screen, and the TA will
say, did you do show?

500
00:37:21 --> 00:37:25
And you'll go -- but it
happens to all of us.

501
00:37:25 --> 00:37:27
All right, so let's try it.


502
00:37:27 --> 00:37:34
See what we get.


503
00:37:34 --> 00:37:41
So sure enough, it's plotted
the values 1, 2, 3, 4, and 5,

504
00:37:41 --> 00:37:44
6, 7, 8, on the x- and y- axis.


505
00:37:44 --> 00:37:49
Two things I want
you to notice here.

506
00:37:49 --> 00:37:56
One Is, that both plots showed
up on the same figure.

507
00:37:56 --> 00:37:58
Which is sometimes what
you want, and sometimes

508
00:37:58 --> 00:37:59
not what you want.


509
00:37:59 --> 00:38:02
You'll notice that also
happened with the random walk

510
00:38:02 --> 00:38:05
we looked at, where when I
plotted five different walks

511
00:38:05 --> 00:38:08
for Homer they all showed up
superimposed on top

512
00:38:08 --> 00:38:12
of one another.


513
00:38:12 --> 00:38:16
The other thing I want you
to notice, is the x-axis

514
00:38:16 --> 00:38:19
runs from 0 to 3.


515
00:38:19 --> 00:38:22
So you might have kind of
thought, that what we

516
00:38:22 --> 00:38:27
would see is a 45 degree
angle on these things.

517
00:38:27 --> 00:38:32
But of course, Python, when
not instructed otherwise,

518
00:38:32 --> 00:38:35
always starts at zero.


519
00:38:35 --> 00:38:43
Since when we called plot, I
gave it only the y-values, it

520
00:38:43 --> 00:38:48
used default values for x.


521
00:38:48 --> 00:38:53
It was smart enough to say,
since we have four y-values, we

522
00:38:53 --> 00:38:57
should need four x-values, and
I'll choose the integers 0,

523
00:38:57 --> 00:39:04
1, 2, 3 as those values.


524
00:39:04 --> 00:39:08
Now you don't have to do that.


525
00:39:08 --> 00:39:18
We could do this instead.


526
00:39:18 --> 00:39:20
Let's try this one.


527
00:39:20 --> 00:39:26
What did I just do?


528
00:39:26 --> 00:39:31
Let's comment these two out,
if we could only get there.

529
00:39:31 --> 00:39:42
This is highly annoying.


530
00:39:42 --> 00:39:46
Let's hope it doesn't tell
me that I have to -- All

531
00:39:46 --> 00:39:50
right, so let's go here.


532
00:39:50 --> 00:39:58
We'll get rid of those guys,
and we'll try this one.

533
00:39:58 --> 00:40:17
We'll plot 1, 2, 3, 4
against 1, 4, 9, 16.

534
00:40:17 --> 00:40:19
OK?


535
00:40:19 --> 00:40:25
So now, it's using 1, 2,
3, 4 as the x-axis, and

536
00:40:25 --> 00:40:28
the y-axis I gave it.


537
00:40:28 --> 00:40:31
First x then y.


538
00:40:31 --> 00:40:34
Now it looks a little funny,
right, you might have not

539
00:40:34 --> 00:40:36
expected it to look like this.


540
00:40:36 --> 00:40:40
You'll notice they're these
little inflection points here.

541
00:40:40 --> 00:40:43
Well, because what it's really
doing is, I gave it a small

542
00:40:43 --> 00:40:46
number of points, only four.


543
00:40:46 --> 00:40:51
It's found those four points,
and it's connected them, each

544
00:40:51 --> 00:40:55
point by a straight line.


545
00:40:55 --> 00:40:59
And since the points are kind
of spread out, the line

546
00:40:59 --> 00:41:03
has little bumps in it.


547
00:41:03 --> 00:41:05
That makes sense to everyone?


548
00:41:05 --> 00:41:09
Now, it's often deceptive to
plot things this way, where you

549
00:41:09 --> 00:41:12
think you have a continuous
function when in fact

550
00:41:12 --> 00:41:17
you just have a few
miscellaneous points.

551
00:41:17 --> 00:41:22
So let's look at
another example.

552
00:41:22 --> 00:41:27
Here, what I'm going to do,
is I've called figure, and

553
00:41:27 --> 00:41:34
remember, this is Pylab
dot figure, which says,

554
00:41:34 --> 00:41:36
create a new figure.


555
00:41:36 --> 00:41:40
So instead of putting this new
curve on the same figure as the

556
00:41:40 --> 00:41:44
old curve, start a new one.


557
00:41:44 --> 00:41:48
And furthermore, I've
got this obscure little

558
00:41:48 --> 00:41:50
thing at the end of it.


559
00:41:50 --> 00:41:54
After you give it the x- and y-
values, you can give it some

560
00:41:54 --> 00:41:59
instructions about how you want
to plot points, or

561
00:41:59 --> 00:42:00
anything else.


562
00:42:00 --> 00:42:05
In this case, what this little
string says is, each point

563
00:42:05 --> 00:42:12
should be represented as a
red o. r for red, o for o.

564
00:42:12 --> 00:42:16
I'm not asking you to remember
this, what you will discover,

565
00:42:16 --> 00:42:21
the good news is there's very
good documentation on this.

566
00:42:21 --> 00:42:27
And so you'll find in the
reading of pointer to plots,

567
00:42:27 --> 00:42:30
and it will tell you everything
you need to know, all of the

568
00:42:30 --> 00:42:34
wizardry and the magic you can
put in these strings that

569
00:42:34 --> 00:42:36
tell you how to do things.


570
00:42:36 --> 00:42:41
These are basically the same
strings borrowed from Matlab.

571
00:42:41 --> 00:42:48
And now if we run it.


572
00:42:48 --> 00:42:53
Figure one is the same
figure we saw before.

573
00:42:53 --> 00:42:58
But figure two has not
connected the dots, not drawn a

574
00:42:58 --> 00:43:05
line, it's actually planted
each, or plotted, excuse me,

575
00:43:05 --> 00:43:11
each point as a red circle.


576
00:43:11 --> 00:43:15
Now when I look at this,
there's something that's not

577
00:43:15 --> 00:43:18
very pleasing about this.


578
00:43:18 --> 00:43:23
That in particular, I know
I plotted four points, but

579
00:43:23 --> 00:43:30
it a quick glance it looks
like they're only three.

580
00:43:30 --> 00:43:36
And that's because it's taking
this fourth point and stuck it

581
00:43:36 --> 00:43:38
way up there in the corner
where I missed it.

582
00:43:38 --> 00:43:40
It's there.


583
00:43:40 --> 00:43:44
But it's so close to the
edge of the graph that

584
00:43:44 --> 00:43:49
it's kind of hard to see.


585
00:43:49 --> 00:44:09
So I can fix that by executing
the command axis, which tells

586
00:44:09 --> 00:44:13
it how far I want it to be.


587
00:44:13 --> 00:44:17
And this says, I want 1 axis
to go from 0 to 6, and

588
00:44:17 --> 00:44:22
the other 0 to 20.


589
00:44:22 --> 00:44:26
We'll do that, and also to
avoid boring you, we'll do

590
00:44:26 --> 00:44:29
more at the same time.


591
00:44:29 --> 00:44:36
We'll put some labels
on these things.

592
00:44:36 --> 00:44:38
I'm going to put that the title
of the graph is going to be

593
00:44:38 --> 00:44:45
earnings, and that the x-axis
will be labelled days, and

594
00:44:45 --> 00:44:48
the y-axis will be
labelled dollars.

595
00:44:48 --> 00:44:55
So earnings dollars
against days.

596
00:44:55 --> 00:45:03
OK, now let's see what
happens when we do this.

597
00:45:03 --> 00:45:07
Well, we get the same ugly
figure one as before, and now

598
00:45:07 --> 00:45:12
you can see figure two I've
moved the axes so that my graph

599
00:45:12 --> 00:45:15
will show up in the middle
rather than at the edges, and

600
00:45:15 --> 00:45:17
therefore easier to read.


601
00:45:17 --> 00:45:22
I put a title in the top, and
I put labels on the axes.

602
00:45:22 --> 00:45:26
Every graph that I ask you to
do this course, I want you to

603
00:45:26 --> 00:45:29
put a title on it and to label
your axes so we know

604
00:45:29 --> 00:45:32
what we're reading.


605
00:45:32 --> 00:45:36
Again, nothing very deep here,
this is really just syntax,

606
00:45:36 --> 00:45:45
just to give you an idea of the
sorts of things you can do.

607
00:45:45 --> 00:45:47
All right. now we get
to something a little

608
00:45:47 --> 00:45:52
bit more interesting.


609
00:45:52 --> 00:46:02
Let's look at this code here.


610
00:46:02 --> 00:46:07
So far, what I've been passing
to the plot function for the

611
00:46:07 --> 00:46:13
x- and y- values are lists.


612
00:46:13 --> 00:46:22
In fact, what Pylab uses is
something it gets from NumPy

613
00:46:22 --> 00:46:33
which are not lists really,
but what it calls arrays.

614
00:46:33 --> 00:46:39
Now, truth be told, most
programming languages use

615
00:46:39 --> 00:46:42
array to mean something
quite different.

616
00:46:42 --> 00:46:53
But, in NumPy an array
is basically a matrix.

617
00:46:53 --> 00:46:56
On which we can do some
interesting things.

618
00:46:56 --> 00:47:05
So for example, when I say
x-axes equals array 1, 2, 3, 4.

619
00:47:05 --> 00:47:09
Array is a type, so array
applied to the list is just

620
00:47:09 --> 00:47:11
like applying float to an int.


621
00:47:11 --> 00:47:14
If I apply float to an
int, it turns it into a

622
00:47:14 --> 00:47:15
floating point number.


623
00:47:15 --> 00:47:22
If I apply array to a list,
it turns it into an array.

624
00:47:22 --> 00:47:25
Once it's an array, as we'll
see, we can do some very

625
00:47:25 --> 00:47:28
interesting things with it.


626
00:47:28 --> 00:47:34
Now, in addition to getting an
array by coercing a the list,

627
00:47:34 --> 00:47:36
which is probably the most
common way to get

628
00:47:36 --> 00:47:37
it, by the way.


629
00:47:37 --> 00:47:40
Because you build up a list in
simulations of the sort we

630
00:47:40 --> 00:47:43
looked at, and then you might
want to change it to an

631
00:47:43 --> 00:47:46
array to perform some
operations on it.

632
00:47:46 --> 00:47:50
You can get an array
directly with aRange.

633
00:47:50 --> 00:47:52
This is just like the range
function we've been using all

634
00:47:52 --> 00:47:56
term, but whereas the range
function gives you a list of

635
00:47:56 --> 00:48:04
ints, this gives you
an array of ints.

636
00:48:04 --> 00:48:08
But the nice thing about an
array is, I can perform

637
00:48:08 --> 00:48:13
operations on it like this.


638
00:48:13 --> 00:48:17
So if I say y-axis equals
x-axis raised to the third

639
00:48:17 --> 00:48:22
power, that's something
I can't do with a list.

640
00:48:22 --> 00:48:24
I get an error message if
I try that with a list.

641
00:48:24 --> 00:48:29
What that will do is, will
point-wise, take each element

642
00:48:29 --> 00:48:34
in the array and cube it.


643
00:48:34 --> 00:48:38
So the nice thing about arrays
is, you can use them to do the

644
00:48:38 --> 00:48:41
kinds of things you, if you
ever took a linear algebra

645
00:48:41 --> 00:48:43
course, you learned
about doing.

646
00:48:43 --> 00:48:46
You can multiply an array times
an array, You can multiply

647
00:48:46 --> 00:48:49
an array times an integer.


648
00:48:49 --> 00:48:53
And sometimes that's a very
convenient thing to do.

649
00:48:53 --> 00:48:55
It's not what this course
is about, I don't

650
00:48:55 --> 00:48:56
want to emphasize it.


651
00:48:56 --> 00:49:00
I just want you to know it's
there so if in some subsequent

652
00:49:00 --> 00:49:06
life you want to do more
complicated manipulations,

653
00:49:06 --> 00:49:09
you'll know that
that's possible.

654
00:49:09 --> 00:49:27
So let's run this and
see what we get.

655
00:49:27 --> 00:49:31
So the first thing to look
at, is we'll ignore the

656
00:49:31 --> 00:49:36
figure for the moment.


657
00:49:36 --> 00:49:42
And we've seen that when I
printed test and I printed

658
00:49:42 --> 00:49:48
x-axis, they look the
same, they are the same.

659
00:49:48 --> 00:49:51
And in fact, I can do this
interesting thing now.

660
00:49:51 --> 00:49:54
Print test double
equals x-axis.

661
00:49:54 --> 00:49:56
You might have thought
that would return a

662
00:49:56 --> 00:50:00
single value, true.


663
00:50:00 --> 00:50:04
Instead it returns a list,
where it's done a point-wise

664
00:50:04 --> 00:50:08
comparison of each element.


665
00:50:08 --> 00:50:13
So when we deal with arrays,
they don't behave like lists.

666
00:50:13 --> 00:50:15
And you can imagine that it
might be very convenient

667
00:50:15 --> 00:50:17
to be able to do this.


668
00:50:17 --> 00:50:21
Answers the question, are
all the elements the same?

669
00:50:21 --> 00:50:23
Or which ones are the same?


670
00:50:23 --> 00:50:27
So you can imagine doing some
very clever things with these.

671
00:50:27 --> 00:50:32
And certainly, if you can
convert problems to vectors of

672
00:50:32 --> 00:50:37
this sort, you can really
perform what's almost magical.

673
00:50:37 --> 00:50:44
And then when we look at the
figure, which should be tucked

674
00:50:44 --> 00:50:51
away somewhere here, what
did I do with the figure?

675
00:50:51 --> 00:50:57
Did I make it go away?


676
00:50:57 --> 00:51:04
Well, I think I did one
of those ugly things and

677
00:51:04 --> 00:51:07
made it go away again.


678
00:51:07 --> 00:51:08
Oh, no, there it is.


679
00:51:08 --> 00:51:10
All right.


680
00:51:10 --> 00:51:14
And sure enough here,
we're plotting a cubic.

681
00:51:14 --> 00:51:16
All right.


682
00:51:16 --> 00:51:21
Nothing very important to
observe about any of that,

683
00:51:21 --> 00:51:23
other than that arrays are
really quite interesting

684
00:51:23 --> 00:51:26
and can be very valuable.


685
00:51:26 --> 00:51:31
Finally, the thing I want to
show you is that there are a

686
00:51:31 --> 00:51:39
lot of things we can do
that are more interesting

687
00:51:39 --> 00:51:46
than what we've done.


688
00:51:46 --> 00:51:49
So now I'm going to use that
random, which I brought in

689
00:51:49 --> 00:51:57
before, and show you that we
can plot things other

690
00:51:57 --> 00:52:01
than simply curves.


691
00:52:01 --> 00:52:04
In this case, I'm going
to plot a histogram.

692
00:52:04 --> 00:52:07
And what this histogram is
going to do is, I'm going to

693
00:52:07 --> 00:52:12
throw a pair of dice a large
number of times, and add up

694
00:52:12 --> 00:52:16
the sum, and see what I get.


695
00:52:16 --> 00:52:20
So, for die values equals 1
through 6, for i in range

696
00:52:20 --> 00:52:22
10,000, a lot of dice.


697
00:52:22 --> 00:52:26
I'm just going to append random
choice, we've seen this before,

698
00:52:26 --> 00:52:30
of the two dice, and their sum.


699
00:52:30 --> 00:52:33
And then I'm going to plot a
histogram, Pylab dot hist

700
00:52:33 --> 00:52:42
instead of plot, and we'll get
something quite different.

701
00:52:42 --> 00:52:47
A nice little histogram showing
me the values I get, and we

702
00:52:47 --> 00:52:50
will come back to this later
and talk about why this is

703
00:52:50 --> 00:52:52
called a normal distribution.


704
00:52:52 --> 00:52:53