1 00:00:04,760 --> 00:00:10,780 The expectation of radar allows us to calculate what the main or average or the expected value is or 2 00:00:10,780 --> 00:00:16,070 a random variable, if it is tested over a large number of experiments, we can see why this would be 3 00:00:16,070 --> 00:00:17,000 useful later on. 4 00:00:17,030 --> 00:00:22,970 But for now, let's suppose that we have an experiment with a discrete random variable X and we're going 5 00:00:22,970 --> 00:00:25,340 to test this variable in times. 6 00:00:25,700 --> 00:00:32,360 Every time we do this, we observe the value of a one and one times the value of a two and two times 7 00:00:32,540 --> 00:00:39,920 and so forth, all the way up to the value of AM occurring and M times the expected value of this random 8 00:00:39,920 --> 00:00:43,310 variable X can be computed using this equation shown below. 9 00:00:44,300 --> 00:00:49,970 So this equation basically sums up the expected value times, the number of times that happens over 10 00:00:49,970 --> 00:00:54,560 the whole sample set and divided by the number of tests to get the expected value. 11 00:00:55,010 --> 00:00:57,370 So now let's use a Disick sample again. 12 00:00:57,380 --> 00:01:04,100 And suppose we roll the dice and times and we count and we count the number of times each number is 13 00:01:04,110 --> 00:01:04,610 shown. 14 00:01:05,270 --> 00:01:07,700 We should end up with a table that looks like this. 15 00:01:08,990 --> 00:01:14,450 So here on this side here, we have the different DI numbers from one to six and from all the different 16 00:01:14,450 --> 00:01:17,650 tests, the 20 tests, this is how many times the one is shown. 17 00:01:17,660 --> 00:01:20,870 This is how many times of the two is shown and so forth. 18 00:01:22,070 --> 00:01:27,830 Now we can use this type of data and the expectation equation up here to calculate what the expected 19 00:01:27,830 --> 00:01:28,540 value is. 20 00:01:29,060 --> 00:01:34,460 So we substitute the numbers into equation with the equation that looks like this segment C, we have 21 00:01:34,640 --> 00:01:35,060 one error. 22 00:01:35,060 --> 00:01:42,950 And so one of the 20 we have the value DI one and we occur twice the value of two occurs four times 23 00:01:43,100 --> 00:01:47,900 and so forth, all the way up to the value of six, which occurs four times as well. 24 00:01:48,500 --> 00:01:52,880 So if we do this equation and we calculate that the expected value is three point eighty five. 25 00:01:53,670 --> 00:01:59,960 Now you may notice the value three point eighty five is not actually any of the expected outcomes in 26 00:02:00,200 --> 00:02:01,100 this great set. 27 00:02:01,700 --> 00:02:05,750 But the expected value does not actually have to be in in the set. 28 00:02:05,780 --> 00:02:09,810 It's just the overall average number from all the experiments. 29 00:02:10,160 --> 00:02:12,320 So let's take this example to the extreme. 30 00:02:12,530 --> 00:02:15,760 Let's assume that we spend the rest of our lives rolling the dice. 31 00:02:16,100 --> 00:02:18,990 What is the actual expected value of the DI then? 32 00:02:19,310 --> 00:02:23,190 Well, we know that the probability of rolling in any number is one of the six. 33 00:02:23,210 --> 00:02:29,870 And from a basic probability, we know so that if we roll the dice and times we expect to see and divided 34 00:02:29,870 --> 00:02:31,270 by six of each number. 35 00:02:31,460 --> 00:02:35,210 So in over six ones and of a six or two's and so forth. 36 00:02:35,900 --> 00:02:39,860 So we can use this and we can write it into the expected value equation. 37 00:02:40,460 --> 00:02:42,110 So we come up with an equation like this. 38 00:02:42,140 --> 00:02:50,270 So for if we run the experiment in times, we should expect in over six ones and over six of twos and 39 00:02:50,270 --> 00:02:50,860 so forth. 40 00:02:51,620 --> 00:02:58,480 And then if we run this experiment and let intends towards infinity, so if we just keep rolling dice 41 00:02:58,490 --> 00:03:04,250 over and over and over again, we can actually calculate the the expected value is three point five. 42 00:03:04,700 --> 00:03:11,520 So we can see that the experiment results are fairly close to what the actual mathematical results are. 43 00:03:12,380 --> 00:03:16,910 So for discrete random variable, we can use the equation that we've shown before. 44 00:03:17,420 --> 00:03:22,910 So now let's extend this concept to continuous random variables and the probability density function. 45 00:03:23,660 --> 00:03:29,900 We can use a probability density function if x to calculate the expected value using this equation here. 46 00:03:30,290 --> 00:03:31,250 So it's very similar. 47 00:03:31,250 --> 00:03:32,510 It's over a summation. 48 00:03:32,510 --> 00:03:36,920 We're using the interval between the two absolute limits of negative energy, infinity. 49 00:03:37,580 --> 00:03:43,970 And we have the value of the random variable X, which is this replaces the eye here and we actually 50 00:03:43,970 --> 00:03:47,180 have the number of times it occurs and it's just going to be the probability. 51 00:03:47,220 --> 00:03:51,950 So you can see that the integral of F of X, the X, that's the area on the curve. 52 00:03:51,950 --> 00:03:53,210 So that's going to be the probability. 53 00:03:54,110 --> 00:03:59,450 So this equation here is simply calculating the main value that we can expect of the random variable 54 00:03:59,460 --> 00:04:06,320 if we test every single possibility, infinite times, the expected value or main value of the random 55 00:04:06,320 --> 00:04:11,350 variable X can be written simplistically using a double annotation. 56 00:04:11,360 --> 00:04:16,170 So we have the random verbal X bar or we could have the little X bar. 57 00:04:16,580 --> 00:04:22,220 So this is different ways of writing that we want to use the mean or the expected value of the random 58 00:04:22,220 --> 00:04:22,730 variable. 59 00:04:23,330 --> 00:04:28,490 Suppose that we ran an experiment on a random variable with a given probability density function multiple 60 00:04:28,490 --> 00:04:34,460 times and a certain random number shows up more times than any other random number from the set. 61 00:04:34,850 --> 00:04:40,370 It just must mean that this number has a higher probability in a density function and the other the 62 00:04:40,370 --> 00:04:41,690 reverse is also true. 63 00:04:41,690 --> 00:04:47,150 If a certain number has a higher probability than the rest, then it's more likely to show up for any 64 00:04:47,150 --> 00:04:47,990 given experiment. 65 00:04:48,680 --> 00:04:52,460 So we can use the expectation of radar to try to quantify this.