1 00:00:00,000 --> 00:00:00,550 2 00:00:00,550 --> 00:00:02,250 Welcome back guys. 3 00:00:02,250 --> 00:00:05,080 Today we're going to work on a problem that tests your 4 00:00:05,080 --> 00:00:07,550 knowledge of joint PMFs. 5 00:00:07,550 --> 00:00:09,620 And we're also going to get some practice computing 6 00:00:09,620 --> 00:00:13,200 conditional expectations and conditional variances. 7 00:00:13,200 --> 00:00:15,740 So in this problem, we are given a set of 8 00:00:15,740 --> 00:00:17,610 points in the xy plane. 9 00:00:17,610 --> 00:00:20,290 And we're told that these points are equally likely. 10 00:00:20,290 --> 00:00:21,210 So there's eight of them. 11 00:00:21,210 --> 00:00:22,410 And each point has a 12 00:00:22,410 --> 00:00:25,500 probability of 1/8 of occurring. 13 00:00:25,500 --> 00:00:27,900 And we're also given this list of questions. 14 00:00:27,900 --> 00:00:30,670 And we're going to work through them together. 15 00:00:30,670 --> 00:00:34,590 So in part a, we are asked to find the values of x that 16 00:00:34,590 --> 00:00:39,010 maximize the conditional expectation of y given x. 17 00:00:39,010 --> 00:00:42,300 So jumping right in, this is the quantity 18 00:00:42,300 --> 00:00:45,190 we're interested in. 19 00:00:45,190 --> 00:00:50,610 And so this quantity is a function of x. 20 00:00:50,610 --> 00:00:53,430 You plug-in various values of x. 21 00:00:53,430 --> 00:00:56,630 And then this will spit out a scalar value. 22 00:00:56,630 --> 00:00:59,970 And that value will correspond to the conditional expectation 23 00:00:59,970 --> 00:01:04,640 of y conditioned on the value of x that you put in. 24 00:01:04,640 --> 00:01:09,690 So let's see, when x is equal to 0, for instance, let's 25 00:01:09,690 --> 00:01:11,410 figure out what this value is. 26 00:01:11,410 --> 00:01:15,790 Well, when x is equal to 0 we're living in a world, 27 00:01:15,790 --> 00:01:17,660 essentially, on this line. 28 00:01:17,660 --> 00:01:19,520 So that means that only these two 29 00:01:19,520 --> 00:01:20,730 points could have occurred. 30 00:01:20,730 --> 00:01:25,550 And in particular, y can only take on the values of 1 and 3. 31 00:01:25,550 --> 00:01:28,580 Now, since all these points in the unconditional universe 32 00:01:28,580 --> 00:01:31,740 were equally likely, in the conditional universe they will 33 00:01:31,740 --> 00:01:33,530 still be equally likely. 34 00:01:33,530 --> 00:01:35,620 So this happens with probability 1/2. 35 00:01:35,620 --> 00:01:38,640 And this happens with probability 1/2. 36 00:01:38,640 --> 00:01:45,960 And therefore, the expectation would just be 3/2 plus 1/2 37 00:01:45,960 --> 00:01:49,380 which is 4/2, or 2. 38 00:01:49,380 --> 00:01:51,700 But a much faster way of seeing this-- and it's the 39 00:01:51,700 --> 00:01:54,300 strategy that I'm going to use for the rest of the problem-- 40 00:01:54,300 --> 00:01:56,720 is to remember that expectation acts 41 00:01:56,720 --> 00:01:58,780 like center of mass. 42 00:01:58,780 --> 00:02:01,490 So the center of mass, when these two points are equally 43 00:02:01,490 --> 00:02:05,010 likely, is just the midpoint, which of course is 2. 44 00:02:05,010 --> 00:02:09,400 So we're going to use that intuition on the other ones. 45 00:02:09,400 --> 00:02:12,730 So I'm skipping to x is equal to 2 because 1 46 00:02:12,730 --> 00:02:14,230 and 3 are not possible. 47 00:02:14,230 --> 00:02:17,000 So when x is equal to 2, y can only take on the 48 00:02:17,000 --> 00:02:18,490 values of 1 or 2. 49 00:02:18,490 --> 00:02:20,240 Again, they're equally likely. 50 00:02:20,240 --> 00:02:22,390 So the center of mass is in the middle which happens at 51 00:02:22,390 --> 00:02:24,000 1.5 or 3/2. 52 00:02:24,000 --> 00:02:26,540 53 00:02:26,540 --> 00:02:29,230 Similarly, x is equal to 4. 54 00:02:29,230 --> 00:02:31,736 We're living in this conditional universe, where y 55 00:02:31,736 --> 00:02:34,380 can take on of these four points with 56 00:02:34,380 --> 00:02:36,870 probability 1/4 each. 57 00:02:36,870 --> 00:02:39,920 And so again, we expect the center of mass to 58 00:02:39,920 --> 00:02:42,850 be at 1.5 or 3/2. 59 00:02:42,850 --> 00:02:45,555 And this quantity is undefined otherwise. 60 00:02:45,555 --> 00:02:48,650 61 00:02:48,650 --> 00:02:50,540 OK, so we're almost done. 62 00:02:50,540 --> 00:02:55,240 Now we just need to find which value of x maximizes this. 63 00:02:55,240 --> 00:02:58,460 Well, let's see, 2 is the biggest quantity out of all of 64 00:02:58,460 --> 00:02:59,630 these numbers. 65 00:02:59,630 --> 00:03:01,320 So the maximum is 2. 66 00:03:01,320 --> 00:03:04,090 And it occurs when x is equal to 0. 67 00:03:04,090 --> 00:03:05,860 So we come over here. 68 00:03:05,860 --> 00:03:07,400 And we found our answer. 69 00:03:07,400 --> 00:03:10,780 x is equal to 0 is the value, which maximizes the 70 00:03:10,780 --> 00:03:13,930 conditional expectation of y given x. 71 00:03:13,930 --> 00:03:15,970 So part b is very similar to part a. 72 00:03:15,970 --> 00:03:18,310 But there is slightly more computation involved. 73 00:03:18,310 --> 00:03:21,170 Because now we're dealing with the variance and not an 74 00:03:21,170 --> 00:03:22,395 expectation. 75 00:03:22,395 --> 00:03:28,220 And variance is usually a little bit tougher to compute. 76 00:03:28,220 --> 00:03:30,250 So we're going to start in the same manner. 77 00:03:30,250 --> 00:03:34,490 But I want you guys to see if you can figure out intuitively 78 00:03:34,490 --> 00:03:36,160 what the right value is. 79 00:03:36,160 --> 00:03:38,160 I'm going to do the entire computation now. 80 00:03:38,160 --> 00:03:41,070 And then you can compare whether your intuition matches 81 00:03:41,070 --> 00:03:43,320 with the real results. 82 00:03:43,320 --> 00:03:49,560 So variance of x conditioned on a particular value of y, 83 00:03:49,560 --> 00:03:51,770 this is now a function of y. 84 00:03:51,770 --> 00:03:54,430 For each value of y you plug in you're going to get out a 85 00:03:54,430 --> 00:03:55,500 scalar number. 86 00:03:55,500 --> 00:03:58,680 And that number represents the conditional variance of x when 87 00:03:58,680 --> 00:04:02,040 you condition on the value of y that you plugged in. 88 00:04:02,040 --> 00:04:07,440 So let's see, when y is equal to 0 we have a nice case. 89 00:04:07,440 --> 00:04:11,680 If y is equal to 0 we have no freedom about what x is. 90 00:04:11,680 --> 00:04:13,820 This is the only point that could have occurred. 91 00:04:13,820 --> 00:04:17,279 Therefore, x definitely takes on a value of 4. 92 00:04:17,279 --> 00:04:19,920 And there's no uncertainty left. 93 00:04:19,920 --> 00:04:23,220 So in other words, the variance is 0. 94 00:04:23,220 --> 00:04:30,040 Now, if y is equal to 1, x can take on a value of 0, a value 95 00:04:30,040 --> 00:04:31,990 of 2 or a value of 4. 96 00:04:31,990 --> 00:04:35,220 And these all have the same probability of occurring, of 97 00:04:35,220 --> 00:04:36,240 1/3, 98 00:04:36,240 --> 00:04:38,920 And again, the reasoning behind that is that all eight 99 00:04:38,920 --> 00:04:42,430 points were equally likely in the unconditional universe. 100 00:04:42,430 --> 00:04:46,990 If you condition on y being equal to 1 these outcomes 101 00:04:46,990 --> 00:04:48,970 still have the same relative frequency. 102 00:04:48,970 --> 00:04:51,190 Namely, they're still equally likely. 103 00:04:51,190 --> 00:04:53,020 And since there are three of them they now have a 104 00:04:53,020 --> 00:04:55,730 probability of 1/3 each. 105 00:04:55,730 --> 00:05:01,430 So we're going to go ahead and use a formula that hopefully, 106 00:05:01,430 --> 00:05:04,080 you guys remember. 107 00:05:04,080 --> 00:05:11,010 So in particular, variance is the expectation of x squared 108 00:05:11,010 --> 00:05:15,520 minus the expectation of x all squared, 109 00:05:15,520 --> 00:05:16,920 the whole thing squared. 110 00:05:16,920 --> 00:05:20,380 So let's start by computing this number first. 111 00:05:20,380 --> 00:05:22,770 So conditioned on y is equal to 1-- 112 00:05:22,770 --> 00:05:24,095 so we're in this line-- 113 00:05:24,095 --> 00:05:27,140 the expectation of x is just 2, right? 114 00:05:27,140 --> 00:05:29,660 The same center-of-mass to argument. 115 00:05:29,660 --> 00:05:34,650 So this, we have a minus 2 squared over here. 116 00:05:34,650 --> 00:05:38,450 Now, x squared is only slightly more difficult. 117 00:05:38,450 --> 00:05:41,960 With probability 1/3, x squared will take 118 00:05:41,960 --> 00:05:43,650 on a value of 0. 119 00:05:43,650 --> 00:05:46,510 With probability 1/3, x squared will take 120 00:05:46,510 --> 00:05:47,990 on a value of 4. 121 00:05:47,990 --> 00:05:49,190 I'm just doing 2 squared. 122 00:05:49,190 --> 00:05:53,180 And with probability 1/3, x squared takes on a value of 4 123 00:05:53,180 --> 00:05:55,520 squared or 16. 124 00:05:55,520 --> 00:05:58,960 So writing down when I just said, we have 0 times 1/3 125 00:05:58,960 --> 00:06:00,610 which is 0. 126 00:06:00,610 --> 00:06:04,310 We have 2 squared, which is 4 times 1/3. 127 00:06:04,310 --> 00:06:08,960 And then we have 4 squared, which is 16 times 1/3. 128 00:06:08,960 --> 00:06:11,710 And then we have our minus 4 from before. 129 00:06:11,710 --> 00:06:18,260 So doing this math out, we get, let's see, 20/3 minus 130 00:06:18,260 --> 00:06:23,320 12/3, which is equal to 8/3, or 8/3. 131 00:06:23,320 --> 00:06:26,790 So we'll come back up here and put 8/3. 132 00:06:26,790 --> 00:06:30,270 So I realize I'm going through this pretty quickly. 133 00:06:30,270 --> 00:06:32,280 Hopefully this step didn't confuse you. 134 00:06:32,280 --> 00:06:35,850 Essentially, when I was doing is, if you think of x squared 135 00:06:35,850 --> 00:06:40,180 as a new random variable, x squared, the possible values 136 00:06:40,180 --> 00:06:44,240 that it can take on are 0, 4, and 16 when you're 137 00:06:44,240 --> 00:06:47,380 conditioning on y is equal to 1. 138 00:06:47,380 --> 00:06:51,490 And so I was simply saying that that random variable 139 00:06:51,490 --> 00:06:56,960 takes on those values with equal probability. 140 00:06:56,960 --> 00:06:59,730 So let's move on to the next one. 141 00:06:59,730 --> 00:07:03,370 So if we condition on y is equal to 2 we're going to do a 142 00:07:03,370 --> 00:07:06,030 very similar computation. 143 00:07:06,030 --> 00:07:08,180 Oops, I shouldn't have erased that. 144 00:07:08,180 --> 00:07:10,790 OK, so we're going to use the same formula that we just 145 00:07:10,790 --> 00:07:16,330 used, which is the expectation of x given y is equal to 2. 146 00:07:16,330 --> 00:07:21,090 Sorry, x squared minus the expectation of x conditioned 147 00:07:21,090 --> 00:07:24,040 on y is equal to 2, all squared. 148 00:07:24,040 --> 00:07:26,470 So conditioned on y is equal to 2, the 149 00:07:26,470 --> 00:07:28,350 expectation of x is 3. 150 00:07:28,350 --> 00:07:30,630 Same center of mass argument. 151 00:07:30,630 --> 00:07:33,150 So 3 squared is 9. 152 00:07:33,150 --> 00:07:38,370 And then x squared can take on a value of 4. 153 00:07:38,370 --> 00:07:40,870 Or it can take on a value of 16. 154 00:07:40,870 --> 00:07:43,280 And it does so with equal probability. 155 00:07:43,280 --> 00:07:51,200 So we get 4/2, 4 plus 16 over 2. 156 00:07:51,200 --> 00:07:55,620 So this is 2 plus 8, which is 10, minus 9. 157 00:07:55,620 --> 00:07:56,630 That'll give us 1. 158 00:07:56,630 --> 00:08:00,710 So we get a 1 when y is equal to 2. 159 00:08:00,710 --> 00:08:03,120 And last computation and then we're done. 160 00:08:03,120 --> 00:08:06,070 I'm still recycling the same formula. 161 00:08:06,070 --> 00:08:11,000 But now we're conditioning on y is equal to 3. 162 00:08:11,000 --> 00:08:14,610 And then we'll be done with this problem, I promise. 163 00:08:14,610 --> 00:08:18,620 OK, so when y is equal to 3 x can take on the value of 0. 164 00:08:18,620 --> 00:08:20,710 Or it can take on the value of 4. 165 00:08:20,710 --> 00:08:24,110 Those two points happen with probability 1/2, 1/2. 166 00:08:24,110 --> 00:08:27,790 So the expectation is right in the middle which is 2. 167 00:08:27,790 --> 00:08:30,210 So we get a minus 4. 168 00:08:30,210 --> 00:08:34,460 And similarly, x squared can take on the value of 0. 169 00:08:34,460 --> 00:08:36,830 When x takes on the value of 0-- and that happens with 170 00:08:36,830 --> 00:08:38,530 probability 1/2-- 171 00:08:38,530 --> 00:08:41,735 similarly, x squared can take on the value of 16 when x 172 00:08:41,735 --> 00:08:42,929 takes on the value of 4. 173 00:08:42,929 --> 00:08:45,400 And that happens with probability 1/2. 174 00:08:45,400 --> 00:08:51,540 So we just have 0/2 plus 16/2 minus 4. 175 00:08:51,540 --> 00:08:55,752 And this gives us 8 minus 4, which is simply 4. 176 00:08:55,752 --> 00:08:59,540 So finally, after all that computation, we are done. 177 00:08:59,540 --> 00:09:03,010 We have the conditional variance of x given y. 178 00:09:03,010 --> 00:09:07,070 Again, we're interested in when this value is largest. 179 00:09:07,070 --> 00:09:11,495 And we see that 4 is the biggest value in this column. 180 00:09:11,495 --> 00:09:15,840 And this value occurs when y takes on a value of 3. 181 00:09:15,840 --> 00:09:22,420 So our answer, over here, is y is equal to 3. 182 00:09:22,420 --> 00:09:25,200 All right, so now we're going to switch gears in part c and 183 00:09:25,200 --> 00:09:27,680 d a little bit. 184 00:09:27,680 --> 00:09:29,660 And we're going to be more concerned 185 00:09:29,660 --> 00:09:32,930 with PMFs, et cetera. 186 00:09:32,930 --> 00:09:37,840 So in part c, we're given a random variable called r which 187 00:09:37,840 --> 00:09:41,250 is defined as the minimum of x and y. 188 00:09:41,250 --> 00:09:43,970 So for instance, this is the 0.01. 189 00:09:43,970 --> 00:09:45,710 The minimum of 0 and 1 is 0. 190 00:09:45,710 --> 00:09:50,180 So r would have a value of 0 here. 191 00:09:50,180 --> 00:09:52,250 Now, we can be a little bit smarter about this. 192 00:09:52,250 --> 00:09:56,250 If we plot the line, y is equal to x. 193 00:09:56,250 --> 00:09:57,855 So that looks something like this. 194 00:09:57,855 --> 00:10:00,360 195 00:10:00,360 --> 00:10:06,780 We see that all of the points below this line satisfy y 196 00:10:06,780 --> 00:10:08,920 being less or equal to x. 197 00:10:08,920 --> 00:10:12,640 And all the points above this line have y greater than or 198 00:10:12,640 --> 00:10:14,290 equal to x. 199 00:10:14,290 --> 00:10:18,332 So if y is less than or equal to x, you hopefully agree that 200 00:10:18,332 --> 00:10:21,660 here the min, or r, is equal to y. 201 00:10:21,660 --> 00:10:26,510 But over here, the min, r, is actually equal to x, since x 202 00:10:26,510 --> 00:10:27,440 is always smaller. 203 00:10:27,440 --> 00:10:28,890 So now we can go ahead quickly. 204 00:10:28,890 --> 00:10:31,760 And I'm going to write the value of r next each point 205 00:10:31,760 --> 00:10:33,190 using this rule. 206 00:10:33,190 --> 00:10:36,960 So here, r is the value of y, which is 1. 207 00:10:36,960 --> 00:10:39,270 Here, r is equal to 0. 208 00:10:39,270 --> 00:10:40,410 Here r is 1. 209 00:10:40,410 --> 00:10:42,200 Here r is 2. 210 00:10:42,200 --> 00:10:44,650 Here r is 3. 211 00:10:44,650 --> 00:10:46,600 Over here, r is the value of x. 212 00:10:46,600 --> 00:10:48,980 So r is equal to 0. 213 00:10:48,980 --> 00:10:50,620 And r is equal to 0 here. 214 00:10:50,620 --> 00:10:53,000 And so the only point we didn't handle is the one that 215 00:10:53,000 --> 00:10:54,120 lies on the line. 216 00:10:54,120 --> 00:10:55,750 But in that case it's easy. 217 00:10:55,750 --> 00:10:57,250 Because x is equal to 2. 218 00:10:57,250 --> 00:10:59,100 And y is equal to 2. 219 00:10:59,100 --> 00:11:01,590 So the min is simply 2. 220 00:11:01,590 --> 00:11:05,040 So with this information I claim we're now done. 221 00:11:05,040 --> 00:11:08,690 We can just write down what the PMF of r is. 222 00:11:08,690 --> 00:11:15,225 So in particular, r takes on a value of 0. 223 00:11:15,225 --> 00:11:17,820 224 00:11:17,820 --> 00:11:20,200 When this point happens, this point happens, 225 00:11:20,200 --> 00:11:21,820 or this point happens. 226 00:11:21,820 --> 00:11:22,990 And those collectively have a 227 00:11:22,990 --> 00:11:27,170 probability of 3/8 of occurring. 228 00:11:27,170 --> 00:11:30,420 r can take on a value of 1 when either of these two 229 00:11:30,420 --> 00:11:32,120 points happen. 230 00:11:32,120 --> 00:11:33,830 So that happens with probability 2/8. 231 00:11:33,830 --> 00:11:36,960 232 00:11:36,960 --> 00:11:38,540 r is equal to 2. 233 00:11:38,540 --> 00:11:40,030 This can happen in two ways. 234 00:11:40,030 --> 00:11:42,450 So we get 2/8. 235 00:11:42,450 --> 00:11:45,740 And r equal to 3 can happen in only one way. 236 00:11:45,740 --> 00:11:48,170 So we get 1/8. 237 00:11:48,170 --> 00:11:51,710 Quick sanity check, 3 plus 2 is 5, plus 2 is 238 00:11:51,710 --> 00:11:52,920 7, plus 1 is 8. 239 00:11:52,920 --> 00:11:55,230 So our PMF sums to 1. 240 00:11:55,230 --> 00:11:57,120 And to be complete, we should sketch it. 241 00:11:57,120 --> 00:12:00,220 Because the problem asks us to sketch it. 242 00:12:00,220 --> 00:12:07,760 So we're plotting PR of r, 0, 1, 2, 3. 243 00:12:07,760 --> 00:12:11,575 So here we get, let's see, 1, 2, 3. 244 00:12:11,575 --> 00:12:14,490 For 0 we have 3/8. 245 00:12:14,490 --> 00:12:17,870 For 1 we have 2/8. 246 00:12:17,870 --> 00:12:20,880 For 2 we have 2/8. 247 00:12:20,880 --> 00:12:24,530 And for 3 we have 1/8. 248 00:12:24,530 --> 00:12:30,090 So this is our fully labeled sketch of Pr of r. 249 00:12:30,090 --> 00:12:34,210 And forgive me for erasing so quickly, but you guys can 250 00:12:34,210 --> 00:12:38,280 pause the video, presumably, if you need more time. 251 00:12:38,280 --> 00:12:40,160 Let's move on to part d. 252 00:12:40,160 --> 00:12:43,590 So in part d we're given an event named a, which is the 253 00:12:43,590 --> 00:12:46,890 event that x squared is greater than or equal to y. 254 00:12:46,890 --> 00:12:50,240 And then we're asked to find the expectation of xy in the 255 00:12:50,240 --> 00:12:51,860 unconditional universe. 256 00:12:51,860 --> 00:12:56,240 And then the expectation of x times y conditioned on a. 257 00:12:56,240 --> 00:12:58,320 So let's not worry about the conditioning for now. 258 00:12:58,320 --> 00:13:01,090 Let's just focus on the unconditional 259 00:13:01,090 --> 00:13:03,040 expectation of x times y. 260 00:13:03,040 --> 00:13:04,630 So I'm just going to erase all these r's 261 00:13:04,630 --> 00:13:07,130 so I don't get confused. 262 00:13:07,130 --> 00:13:12,910 But we're going to follow a very similar strategy, which 263 00:13:12,910 --> 00:13:17,770 is at each point I'm going to label what the value of w is. 264 00:13:17,770 --> 00:13:22,840 And we'll find the expectation of w that way. 265 00:13:22,840 --> 00:13:27,130 So let's see, here, we have 4 times 0. 266 00:13:27,130 --> 00:13:29,050 So w is equal to 0. 267 00:13:29,050 --> 00:13:30,440 Here we have 4 times 1. 268 00:13:30,440 --> 00:13:32,340 w is equal to 4. 269 00:13:32,340 --> 00:13:34,530 4 times 2, w is equal to 8. 270 00:13:34,530 --> 00:13:37,882 4 times 3, w is equal to 12. 271 00:13:37,882 --> 00:13:39,730 w is equal to 2. 272 00:13:39,730 --> 00:13:41,980 w is equal to 4. 273 00:13:41,980 --> 00:13:43,610 w is equal to 0. 274 00:13:43,610 --> 00:13:45,430 w is equal to 0. 275 00:13:45,430 --> 00:13:47,610 OK, so that was just algebra. 276 00:13:47,610 --> 00:13:51,050 And now, I claim again, we can just write down what the 277 00:13:51,050 --> 00:13:54,450 expectation of x times y is. 278 00:13:54,450 --> 00:13:57,550 And I'm sorry, I didn't announce my notation. 279 00:13:57,550 --> 00:14:00,560 I should mention that now. 280 00:14:00,560 --> 00:14:04,640 I was defining w to be the random variable x times y. 281 00:14:04,640 --> 00:14:06,910 And that's why I labeled the product of x 282 00:14:06,910 --> 00:14:10,760 times y as w over here. 283 00:14:10,760 --> 00:14:15,330 My apologies about not defining that random variable. 284 00:14:15,330 --> 00:14:21,210 So the expectation of w, well, w takes on a value of 0. 285 00:14:21,210 --> 00:14:23,680 When this happens, this happens or that happens. 286 00:14:23,680 --> 00:14:28,920 And we know that those three points occur 287 00:14:28,920 --> 00:14:31,260 with probability 3/8. 288 00:14:31,260 --> 00:14:35,100 So we have 0 times 3/8. 289 00:14:35,100 --> 00:14:38,080 I'm just using the normal formula for expectation. 290 00:14:38,080 --> 00:14:40,815 w takes on a value of 2 with probability 1/8. 291 00:14:40,815 --> 00:14:43,590 Because this is the lead point in which it 292 00:14:43,590 --> 00:14:46,220 happens, 2 times 1/8. 293 00:14:46,220 --> 00:14:50,770 Plus it can take on the value of 4 with probability 294 00:14:50,770 --> 00:14:54,260 2/8, 4 times 2/8. 295 00:14:54,260 --> 00:14:58,150 And 8, with 1/8 probability. 296 00:14:58,150 --> 00:15:01,740 And similarly, 12 with 1/8 probability. 297 00:15:01,740 --> 00:15:03,150 So this is just algebra. 298 00:15:03,150 --> 00:15:06,674 The numerator sums up to 30. 299 00:15:06,674 --> 00:15:08,445 Yes, that's correct. 300 00:15:08,445 --> 00:15:13,920 So we have 30/8, which is equal to 15/4. 301 00:15:13,920 --> 00:15:18,390 So this is our first answer for part d. 302 00:15:18,390 --> 00:15:20,610 And now we have to do this slightly trickier one, which 303 00:15:20,610 --> 00:15:24,290 is the conditional expectation of x times y, or w 304 00:15:24,290 --> 00:15:26,110 conditioned on a. 305 00:15:26,110 --> 00:15:31,360 So similar to what I did in part c, I'm going to draw the 306 00:15:31,360 --> 00:15:34,510 line y equals x squared. 307 00:15:34,510 --> 00:15:37,870 So y equals x squared is 0 here, 1 here. 308 00:15:37,870 --> 00:15:41,070 And at 2, it should take on a value of 4. 309 00:15:41,070 --> 00:15:45,760 So the curve should look something like this. 310 00:15:45,760 --> 00:15:48,470 This is the line y is equal to x squared. 311 00:15:48,470 --> 00:15:52,410 So we know all the points below this line satisfy y less 312 00:15:52,410 --> 00:15:54,230 than or equal to x squared. 313 00:15:54,230 --> 00:15:57,130 And all the points above this line have y greater than or 314 00:15:57,130 --> 00:15:59,170 equal to x squared. 315 00:15:59,170 --> 00:16:02,870 And a is y less than or equal to x squared. 316 00:16:02,870 --> 00:16:07,090 So we are in the conditional universe where only points 317 00:16:07,090 --> 00:16:09,390 below this line can happen. 318 00:16:09,390 --> 00:16:11,240 So that one, that one, that one, that one, that 319 00:16:11,240 --> 00:16:12,190 one and that one. 320 00:16:12,190 --> 00:16:13,790 So there are six of them. 321 00:16:13,790 --> 00:16:15,930 And again, in the unconditional world, all of 322 00:16:15,930 --> 00:16:17,190 the points were equally likely. 323 00:16:17,190 --> 00:16:20,040 So in the conditional world these six points are still 324 00:16:20,040 --> 00:16:21,040 equally likely. 325 00:16:21,040 --> 00:16:24,630 So they each happen with probability 1/6. 326 00:16:24,630 --> 00:16:33,310 So in this case, the expectation of w is simply 2 327 00:16:33,310 --> 00:16:38,050 times 1/6 plus 0 times 1/6. 328 00:16:38,050 --> 00:16:38,930 But that's 0. 329 00:16:38,930 --> 00:16:41,260 So I'm not going to write it. 330 00:16:41,260 --> 00:16:52,250 4 times 2/6 plus 4 times 2/6 plus 8 times 1/6, plus 12 331 00:16:52,250 --> 00:16:54,820 times 1 over 6. 332 00:16:54,820 --> 00:16:56,650 And again, the numerator summed to 30. 333 00:16:56,650 --> 00:16:58,780 But this time our denominator is 6. 334 00:16:58,780 --> 00:17:01,210 So this is simply 5. 335 00:17:01,210 --> 00:17:04,230 So we have, actually, finished the problem. 336 00:17:04,230 --> 00:17:07,940 Because we've computed this value and this value. 337 00:17:07,940 --> 00:17:12,369 And so the important takeaways of this problem are, 338 00:17:12,369 --> 00:17:16,089 essentially, honestly, just to get you comfortable with 339 00:17:16,089 --> 00:17:19,410 computing things involving joint PMFs. 340 00:17:19,410 --> 00:17:23,109 We talked a lot about finding expectations quickly by 341 00:17:23,109 --> 00:17:25,180 thinking about center of mass and the 342 00:17:25,180 --> 00:17:27,030 geometry of the problem. 343 00:17:27,030 --> 00:17:31,461 We've got practice computing conditional variances. 344 00:17:31,461 --> 00:17:34,330 And we did some derived distributions. 345 00:17:34,330 --> 00:17:36,170 And we'll do a lot more of those later. 346 00:17:36,170 --> 00:17:37,420