1 00:00:01,040 --> 00:00:05,610 We now develop a methodology for finding the PDF of the sum 2 00:00:05,610 --> 00:00:08,760 of two independent random variables, when these random 3 00:00:08,760 --> 00:00:11,710 variables are continuous with known PDFs. 4 00:00:11,710 --> 00:00:14,780 So in that case, Z will also be continuous and 5 00:00:14,780 --> 00:00:16,530 so will have a PDF. 6 00:00:16,530 --> 00:00:19,430 The development is quite analogous to the one for the 7 00:00:19,430 --> 00:00:20,340 discrete case. 8 00:00:20,340 --> 00:00:22,100 And in the discrete case, we obtained 9 00:00:22,100 --> 00:00:24,090 this convolution formula. 10 00:00:24,090 --> 00:00:27,980 This convolution formula corresponds to a summation 11 00:00:27,980 --> 00:00:33,050 over all ways that a certain sum can be realized. 12 00:00:33,050 --> 00:00:36,520 In this picture, these are all the ways that the sum of 3 can 13 00:00:36,520 --> 00:00:37,930 be realized. 14 00:00:37,930 --> 00:00:40,400 In the continuous case, the different ways that the 15 00:00:40,400 --> 00:00:44,340 constant sum can be realized corresponds to a line. 16 00:00:44,340 --> 00:00:50,370 So this is a line in which X plus Y is equal to a constant. 17 00:00:50,370 --> 00:00:54,890 And we need to somehow add over all the possible ways 18 00:00:54,890 --> 00:00:58,330 that the sum can be obtained, add over all the 19 00:00:58,330 --> 00:00:59,850 points on this line. 20 00:00:59,850 --> 00:01:03,970 Now, when we're summing over all the points of the line we 21 00:01:03,970 --> 00:01:06,480 really need to employ an integral. 22 00:01:06,480 --> 00:01:08,220 And this leads to the following 23 00:01:08,220 --> 00:01:10,050 guess for the formula. 24 00:01:10,050 --> 00:01:13,840 Instead of having a summation, we will have an integral. 25 00:01:13,840 --> 00:01:21,120 And the integral is over all the X, Y pairs whose sum is a 26 00:01:21,120 --> 00:01:23,510 constant number, little z. 27 00:01:23,510 --> 00:01:25,860 So we have here the family recipe-- 28 00:01:25,860 --> 00:01:30,235 that sums are replaced by integrals and PMFs are 29 00:01:30,235 --> 00:01:31,870 replaced by PDFs. 30 00:01:31,870 --> 00:01:35,030 So this formula is entirely plausible. 31 00:01:35,030 --> 00:01:36,250 And it is called the 32 00:01:36,250 --> 00:01:39,140 continuous convolution formula. 33 00:01:39,140 --> 00:01:42,750 What we want to do next is to actually justify this formula 34 00:01:42,750 --> 00:01:44,770 more rigorously. 35 00:01:44,770 --> 00:01:46,890 We will use the following trick. 36 00:01:46,890 --> 00:01:52,310 We will first condition on the random variable X, taking on a 37 00:01:52,310 --> 00:01:53,850 specific value. 38 00:01:53,850 --> 00:01:58,080 If we do this conditioning, then the random variable Z 39 00:01:58,080 --> 00:02:06,110 becomes little x plus Y. And to make the argument more 40 00:02:06,110 --> 00:02:10,070 transparent, we're going to look first at the special case 41 00:02:10,070 --> 00:02:14,120 where little x is let's say, the number 3. 42 00:02:14,120 --> 00:02:17,400 In which case our random variable Z is 43 00:02:17,400 --> 00:02:20,300 equal to Y plus 3. 44 00:02:20,300 --> 00:02:24,990 Let us now calculate the conditional PDF of Z in a 45 00:02:24,990 --> 00:02:30,020 universe in which we are told that the random variable X 46 00:02:30,020 --> 00:02:33,300 takes on the value of 3. 47 00:02:33,300 --> 00:02:37,440 Now, given that X takes on the value of 3, the random 48 00:02:37,440 --> 00:02:42,570 variable Z is the same as the random variable Y plus 3. 49 00:02:50,200 --> 00:02:53,270 And now we have the conditional PDF of y plus 3 50 00:02:53,270 --> 00:02:54,360 given X. 51 00:02:54,360 --> 00:02:58,120 However, we have assumed that X and Y are independent. 52 00:02:58,120 --> 00:03:02,180 So the conditional PDF is going to be the same as the 53 00:03:02,180 --> 00:03:05,400 unconditional PDF of Y plus 3. 54 00:03:09,210 --> 00:03:11,720 And we obtain this expression. 55 00:03:11,720 --> 00:03:14,030 Now, what is this? 56 00:03:14,030 --> 00:03:19,410 We know the PDF of Y. But now we want the PDF of Y plus 3, 57 00:03:19,410 --> 00:03:23,550 which is a simple version of a linear function of a single 58 00:03:23,550 --> 00:03:28,940 random variable Y. For a linear function of this form, 59 00:03:28,940 --> 00:03:32,070 we have already derived a formula. 60 00:03:32,070 --> 00:03:35,530 In the notation we have used in the past, if we have a 61 00:03:35,530 --> 00:03:42,610 random variable X, and we add the constant to it, the PDF of 62 00:03:42,610 --> 00:03:48,240 the new random variable is the PDF of X but shifted by an 63 00:03:48,240 --> 00:03:51,230 amount equal to b to the right. 64 00:03:51,230 --> 00:03:55,370 And that's what the shifting corresponds to mathematically. 65 00:03:55,370 --> 00:03:57,930 Now, let's us apply this formula to the case 66 00:03:57,930 --> 00:03:59,400 that we have here. 67 00:03:59,400 --> 00:04:02,440 We need to keep track of the different symbols. 68 00:04:02,440 --> 00:04:07,700 So capital Y corresponds to X, b corresponds to 3, little x 69 00:04:07,700 --> 00:04:11,780 corresponds to Z. And by using these correspondences, what we 70 00:04:11,780 --> 00:04:18,329 obtain is f sub Y of this argument, which is Z in our 71 00:04:18,329 --> 00:04:24,100 case minus b, which is 3 in our case. 72 00:04:24,100 --> 00:04:27,990 And this is the final form for the conditional density of Z 73 00:04:27,990 --> 00:04:30,530 given that X takes a specific value. 74 00:04:30,530 --> 00:04:35,140 It's nothing more than the density of Y, but shifted by 3 75 00:04:35,140 --> 00:04:38,100 units to the right. 76 00:04:38,100 --> 00:04:39,840 Let us now generalize this. 77 00:04:39,840 --> 00:04:44,830 Instead of using X equal to 3, let us use a general number. 78 00:04:44,830 --> 00:04:48,390 And this gives us the more general formula, that the 79 00:04:48,390 --> 00:04:53,090 conditional PDF of Z given that X takes on a specific 80 00:04:53,090 --> 00:04:55,460 value is equal to-- 81 00:04:55,460 --> 00:04:58,600 just use little x here instead of 3. 82 00:04:58,600 --> 00:05:00,110 It takes this form. 83 00:05:04,340 --> 00:05:07,510 So we do have now in our hands a formula for the conditional 84 00:05:07,510 --> 00:05:09,850 density of Z given X. 85 00:05:09,850 --> 00:05:15,040 Since we have the conditional, and we also know the PDF of X, 86 00:05:15,040 --> 00:05:20,520 we can use the multiplication rule to find the joint PDF of 87 00:05:20,520 --> 00:05:28,710 X and Z. By the multiplication rule, it is the marginal PDF 88 00:05:28,710 --> 00:05:34,130 of X times the conditional PDF of Z given X, which in our 89 00:05:34,130 --> 00:05:36,530 case takes this particular form. 90 00:05:39,320 --> 00:05:43,830 And now that we have the joint PDF in our hands, we can use 91 00:05:43,830 --> 00:05:46,420 another familiar formula that takes us from the 92 00:05:46,420 --> 00:05:48,060 joint to the marginal. 93 00:05:48,060 --> 00:05:52,010 It would take the joint PDF and integrate with respect to 94 00:05:52,010 --> 00:05:56,690 one argument, we obtain the marginal PDF of the other 95 00:05:56,690 --> 00:05:59,370 random variable. 96 00:05:59,370 --> 00:06:03,860 Using this specific form that we have for the joint PDF in 97 00:06:03,860 --> 00:06:08,320 this formula, we have finally obtained this expression. 98 00:06:08,320 --> 00:06:13,120 This is the integral of the joint PDF of X with Z 99 00:06:13,120 --> 00:06:15,370 integrated over all xs. 100 00:06:15,370 --> 00:06:18,910 And this proves this convolution formula. 101 00:06:18,910 --> 00:06:23,350 In terms of the mechanics of carrying out the calculation 102 00:06:23,350 --> 00:06:26,500 of the convolution, the mechanics are exactly the same 103 00:06:26,500 --> 00:06:28,180 as in the discrete case. 104 00:06:28,180 --> 00:06:31,090 If you want to solve a problem graphically, what you will do 105 00:06:31,090 --> 00:06:36,300 is to take the PDF of Y, flip it horizontally, and then 106 00:06:36,300 --> 00:06:41,070 shift it by an amount of little z, cross multiply 107 00:06:41,070 --> 00:06:43,760 terms, and integrate them out.