1 00:00:00,000 --> 00:00:00,490 2 00:00:00,490 --> 00:00:01,360 Hi. 3 00:00:01,360 --> 00:00:03,990 In this video, we're going to do standard probability 4 00:00:03,990 --> 00:00:07,640 calculations for normal random variables. 5 00:00:07,640 --> 00:00:10,670 We're given that x is standard normal with mean 0 and 6 00:00:10,670 --> 00:00:11,730 variance 1. 7 00:00:11,730 --> 00:00:15,420 And y is normal with mean one and variance 4. 8 00:00:15,420 --> 00:00:18,310 And we're asked for a couple of probabilities. 9 00:00:18,310 --> 00:00:22,400 For the normal CDF, we don't have a closed form expression. 10 00:00:22,400 --> 00:00:26,610 And so people generally tabulate values and for the 11 00:00:26,610 --> 00:00:28,570 standard normal case. 12 00:00:28,570 --> 00:00:34,880 So if we want little x equal to 3.49, we just look for 3.4 13 00:00:34,880 --> 00:00:37,970 along the rows and 0.09 along the columns, and then pick the 14 00:00:37,970 --> 00:00:39,720 value appropriately. 15 00:00:39,720 --> 00:00:43,680 So for part A, we're asked what's the probability that x 16 00:00:43,680 --> 00:00:46,230 is less than equal to 1.5? 17 00:00:46,230 --> 00:00:50,110 That's exactly phi of 1.5 and we can look that up. 18 00:00:50,110 --> 00:00:56,950 1.5 directly and that's 0.9332. 19 00:00:56,950 --> 00:00:59,260 Then were asked, what's the probability that x is less 20 00:00:59,260 --> 00:01:02,250 than equal to negative 1? 21 00:01:02,250 --> 00:01:05,610 Notice that negative values are not on this table. 22 00:01:05,610 --> 00:01:09,800 And the reason that is is because the standard normal is 23 00:01:09,800 --> 00:01:11,140 symmetric around zero. 24 00:01:11,140 --> 00:01:12,780 And we don't really need that. 25 00:01:12,780 --> 00:01:15,490 We just recognize that the area in this region is exactly 26 00:01:15,490 --> 00:01:17,490 the area in this region. 27 00:01:17,490 --> 00:01:20,320 And so that's equal to the probability that x is greater 28 00:01:20,320 --> 00:01:22,920 than equal to 1. 29 00:01:22,920 --> 00:01:25,940 This is equal to 1 minus the probability that x 30 00:01:25,940 --> 00:01:27,920 is less than 1. 31 00:01:27,920 --> 00:01:31,350 And we can put the equal sign in here because x is 32 00:01:31,350 --> 00:01:33,370 continuous, it doesn't matter. 33 00:01:33,370 --> 00:01:35,570 And so we're going to get, this is equal to 1 34 00:01:35,570 --> 00:01:38,020 minus phi of one. 35 00:01:38,020 --> 00:01:40,310 And we can look up phi of 1, which is 36 00:01:40,310 --> 00:01:49,650 1.00, and that's 0.8413. 37 00:01:49,650 --> 00:01:51,330 OK. 38 00:01:51,330 --> 00:01:56,050 For part B, we're asked for this distribution of y 39 00:01:56,050 --> 00:01:58,160 minus 1 over 2. 40 00:01:58,160 --> 00:02:00,650 So any linear function of a normal random 41 00:02:00,650 --> 00:02:02,860 variable is also normal. 42 00:02:02,860 --> 00:02:06,930 And you can see that by using the derived distribution for 43 00:02:06,930 --> 00:02:12,800 linear functions of random variables. 44 00:02:12,800 --> 00:02:15,360 So in this case, we only need to figure out what's the mean 45 00:02:15,360 --> 00:02:18,620 and the variance of this normal random variable. 46 00:02:18,620 --> 00:02:22,880 So the mean in this case, I'm going to write that as y over 47 00:02:22,880 --> 00:02:24,130 2 minus 1/2. 48 00:02:24,130 --> 00:02:27,020 49 00:02:27,020 --> 00:02:28,970 The expectation operator is linear and so 50 00:02:28,970 --> 00:02:30,220 that's going to be-- 51 00:02:30,220 --> 00:02:34,990 52 00:02:34,990 --> 00:02:38,810 and the expectation in this case is 1, so 53 00:02:38,810 --> 00:02:41,310 that's going to be 0. 54 00:02:41,310 --> 00:02:42,560 Now the variance. 55 00:02:42,560 --> 00:02:45,840 56 00:02:45,840 --> 00:02:49,320 For the shift, it doesn't affect the spread. 57 00:02:49,320 --> 00:02:52,530 And so the variance is exactly going to be the same without 58 00:02:52,530 --> 00:02:55,300 the minus 1/2. 59 00:02:55,300 --> 00:02:57,240 And for the constant, you can just pull that 60 00:02:57,240 --> 00:02:58,490 out and square it. 61 00:02:58,490 --> 00:03:01,530 62 00:03:01,530 --> 00:03:04,410 And the variance of y we know is 4. 63 00:03:04,410 --> 00:03:07,670 And so that's 1/4 times 4, that's 1. 64 00:03:07,670 --> 00:03:08,170 OK. 65 00:03:08,170 --> 00:03:11,190 So now we know that y minus 1 over 2 is 66 00:03:11,190 --> 00:03:12,700 actually standard normal. 67 00:03:12,700 --> 00:03:16,740 68 00:03:16,740 --> 00:03:21,840 Actually for any normal random variable, you can follow the 69 00:03:21,840 --> 00:03:22,500 same procedure. 70 00:03:22,500 --> 00:03:24,960 You just subtract its mean, which is 1 in this case. 71 00:03:24,960 --> 00:03:28,050 And divide by its standard deviation and you will get a 72 00:03:28,050 --> 00:03:31,520 standard normal distribution. 73 00:03:31,520 --> 00:03:37,560 All right, so for part C we want the probability that y is 74 00:03:37,560 --> 00:03:42,190 between negative 1 and 1. 75 00:03:42,190 --> 00:03:45,420 So let's try to massage it so that we can use the standard 76 00:03:45,420 --> 00:03:46,980 normal table. 77 00:03:46,980 --> 00:03:49,430 And we already know that this is standard normal, so let's 78 00:03:49,430 --> 00:03:53,210 subtract both sides by negative 1. 79 00:03:53,210 --> 00:04:04,240 80 00:04:04,240 --> 00:04:05,170 And that's equal to-- 81 00:04:05,170 --> 00:04:10,120 I'm going to call this standard normal z, so that's 82 00:04:10,120 --> 00:04:11,600 easier to write. 83 00:04:11,600 --> 00:04:16,100 And that's equal to negative 1 less than equal to z, less 84 00:04:16,100 --> 00:04:17,670 than equal to zero. 85 00:04:17,670 --> 00:04:23,600 So we're looking for this region, 0, 1, negative 1. 86 00:04:23,600 --> 00:04:26,900 87 00:04:26,900 --> 00:04:37,710 So that's just the probability that it's less than zero minus 88 00:04:37,710 --> 00:04:42,840 probability that it's less than negative 1. 89 00:04:42,840 --> 00:04:46,790 Well for a standard normal, half the mass is below zero 90 00:04:46,790 --> 00:04:48,490 and a half the mass is above. 91 00:04:48,490 --> 00:04:51,960 And so that's just going to be 0.5 directly. 92 00:04:51,960 --> 00:04:56,460 And for this, we've already computed this for a standard 93 00:04:56,460 --> 00:04:59,290 normal, which was x in our case. 94 00:04:59,290 --> 00:05:09,868 And that was 1 minus 0.8413. 95 00:05:09,868 --> 00:05:10,870 Done. 96 00:05:10,870 --> 00:05:16,320 So we basically calculated a few standard probabilities for 97 00:05:16,320 --> 00:05:17,560 normal distributions. 98 00:05:17,560 --> 00:05:20,950 And we did that by looking them up from the standard 99 00:05:20,950 --> 00:05:22,200 normal table. 100 00:05:22,200 --> 00:05:24,367