WEBVTT
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As before, we have a red Poisson
process and a green Poisson
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process.
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We merge these two
processes, and we only
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observe the merged process.
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Here's an interesting question.
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This is an arrival of
the merged process.
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Where did it come from?
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Is it red, or is it green?
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We cannot know, but can we tell
what is the probability that it
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came from the red stream?
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The way to answer
this question is
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to look at the table
of all the things
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that can happen during
a little interval
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around that particular time
in which we had an arrival.
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We are told that there was an
arrival at time t or an arrival
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during an interval, a small
interval around time t.
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This means that we're told that
this event here has happened.
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Given this information, what
is the conditional probability
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that actually this
event here occurred?
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This is just the fraction
of this probability divided
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by the total probability
of the conditioning event.
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So the answer is
lambda 1 divided
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by lambda 1 plus lambda 2.
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Does this answer make sense?
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Well, suppose that lambda
1 and lambda 2 were equal.
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In that case, by symmetry,
when an arrival comes,
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it should be equally likely to
have come either from the red
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or from the green stream.
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And this is consistent
with this answer.
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We can reason similarly for a
slightly different question.
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You wait until the kth arrival,
let's say the third arrival.
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Where did that
arrival come from?
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Well, that case,
arrival occurred
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during a specific little time
interval and conditioning
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on it having occurred during
that particular time interval,
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we can then repeat the reasoning
that we had here and argue
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that given that we had an
arrival-- it just happens
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to be the third arrival
during that time interval--
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there's going to be this
particular conditional
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probability that it came
from the red stream.
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So we obtained the
same answer once more.
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Now, this arrival came
from one of the two streams
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with some probabilities.
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This arrival came from
one of the two streams
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with some probabilities.
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Does the origin of this
arrival affect or depend
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on the origin of that arrival?
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Because we have
assumed independence
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across time for each
one of the processes
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that we started with--
and therefore, we also
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have the same thing for the
merged process-- whatever
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has to do with events
during this interval
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is independent
from anything that
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has to do with events
in that interval.
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And because of this, one
could argue formally--
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but hopefully, this
is intuitive enough--
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that the origin of this arrival
and the origin of that arrival
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are independent events.
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And now that we
have this property,
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we can answer questions
such as the following.
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We've had 10 arrivals so far.
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What is the probability that
exactly four out of these 10
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are red?
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Each one of those arrivals has
this probability of being red.
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The origin of different arrivals
are independent of each other.
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So essentially, we're dealing
with 10 Bernoulli trials,
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each of which has two possible
outcomes, red or green,
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and is red with this
particular probability.
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Therefore, the
answer is going to be
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given by the binomial
probabilities, which
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is the probability of having
four successes in 10 trials.
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And we obtain lambda 1 over
lambda 1 plus lambda 2.
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That's the probability of a red
to the number of red arrivals.
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And then the
remaining probability,
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1 minus that, which is
lambda 2 over lambda 1
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plus lambda 2 to the remaining
power, which is equal to 6.
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So to summarize.
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Each one of the arrivals
in the merged process
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has a certain
probability of being
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a red arrival or
a green arrival.
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Which one of the
two is the case?
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We can think of it as an
outcome of Bernoulli trial,
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and the Bernoulli
trials associated
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with different arrivals are
independent of each other
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as a consequence of the
independence of Poisson
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processes across time.