1 00:00:00,120 --> 00:00:04,830 Next, we want to talk about a really important probability distribution because it comes up all the 2 00:00:04,830 --> 00:00:09,330 time in statistics, and that is the continuous uniform distribution. 3 00:00:09,540 --> 00:00:11,220 Now it looks like this. 4 00:00:11,220 --> 00:00:12,420 It's really simple. 5 00:00:12,420 --> 00:00:17,850 This is a simple distribution where the probability density function associated with this distribution 6 00:00:17,850 --> 00:00:25,650 is completely off or at zero for all values of x until we get to a So everything to the left of x equals 7 00:00:25,650 --> 00:00:27,450 a here is it zero. 8 00:00:27,450 --> 00:00:33,930 And then at x equals A, suddenly we see the distribution jump up to this level here. 9 00:00:34,080 --> 00:00:41,160 Then it b it immediately instantaneously drops down back to zero again and remains at zero. 10 00:00:41,190 --> 00:00:45,300 These vertical dashed lines are not part of the distribution. 11 00:00:45,300 --> 00:00:51,330 The reason they're there is to show us that this part of the distribution up here starts exactly at 12 00:00:51,330 --> 00:00:58,710 a and ends exactly at B, But the distribution itself just has this zero value to the left of A, then 13 00:00:58,710 --> 00:01:05,670 this value up here between A and B and then again becomes zero at B and stays at zero for all other 14 00:01:05,670 --> 00:01:07,500 values of X one. 15 00:01:07,500 --> 00:01:13,110 Application of this kind of continuous uniform distribution is this idea of voltage. 16 00:01:13,110 --> 00:01:20,040 Think about turning on a light switch When the light is off, the voltage being supplied is zero, and 17 00:01:20,040 --> 00:01:26,070 then suddenly if we flip the light switch on at a the voltage jumps up to this level here and then if 18 00:01:26,070 --> 00:01:30,180 we turn the light switch off at B, it drops down again to zero. 19 00:01:30,210 --> 00:01:34,880 That's kind of one way to remember this off, on off kind of pattern. 20 00:01:34,890 --> 00:01:41,520 Now, before we go further with this uniform distribution, let's compare and contrast this distribution 21 00:01:41,640 --> 00:01:46,590 with the one that we looked at before when we talked about discrete uniform distributions. 22 00:01:46,590 --> 00:01:53,310 If you remember, this is the probability mass function that is associated with rolling a six sided 23 00:01:53,310 --> 00:01:54,570 die one time. 24 00:01:54,570 --> 00:02:00,030 And this represents the discrete uniform distribution of that die roll. 25 00:02:00,030 --> 00:02:06,780 So when we have a discrete random variable and its probability mass function and we want to sketch its 26 00:02:06,780 --> 00:02:12,720 probability distribution, it's going to look like this where we have these vertical bars or columns, 27 00:02:12,720 --> 00:02:16,860 all of equal height at each of the values that X can take on. 28 00:02:16,860 --> 00:02:23,820 So all of these bars or columns in red that represent the probability are vertical, whereas with a 29 00:02:23,820 --> 00:02:30,270 continuous random variable where we have a continuous uniform distribution, all of the pieces of our 30 00:02:30,270 --> 00:02:33,210 probability density function are horizontal. 31 00:02:33,240 --> 00:02:39,450 This section of the probability density function between x equals zero and x equals a is horizontal. 32 00:02:39,480 --> 00:02:42,420 This piece of it here between A and B is horizontal. 33 00:02:42,420 --> 00:02:47,430 This piece here starting at B and extending to the right of B is also horizontal. 34 00:02:47,430 --> 00:02:52,890 And that makes sense because when we have a continuous random variable, remember we can always define 35 00:02:52,890 --> 00:02:59,880 smaller and smaller increments of this continuous random variable x, so we can define X at A or at 36 00:02:59,880 --> 00:03:05,910 B, but we can also define X halfway between A and B, and then halfway between those values again and 37 00:03:05,910 --> 00:03:06,870 then halfway again. 38 00:03:06,870 --> 00:03:11,310 And we can continue measuring smaller and smaller increments of X. 39 00:03:11,310 --> 00:03:16,980 And so when we're talking about the probability distribution of a continuous random variable like that, 40 00:03:16,980 --> 00:03:22,860 we need to model probability along all of these values of X along this horizontal axis. 41 00:03:22,860 --> 00:03:29,190 And so we have to have these horizontal sections of the graph that can represent the value of the probability 42 00:03:29,190 --> 00:03:36,660 density function F of X at all of these different values of X along the axis, whereas it would never 43 00:03:36,660 --> 00:03:44,160 make sense to sketch a continuous uniform distribution for a discrete random variable because it would 44 00:03:44,160 --> 00:03:49,410 have to look something like this where we're showing that the probability that these values of X occurs 45 00:03:49,410 --> 00:03:56,760 is always 0.167 always consistent, but that only works at this exact value. 46 00:03:56,790 --> 00:04:01,080 X equals one right here at this exact value x equals two. 47 00:04:01,110 --> 00:04:07,890 Here at this exact value x equals three here, it doesn't make sense to show this horizontal section 48 00:04:07,890 --> 00:04:11,640 of graph in between one and two or in between two and three. 49 00:04:11,640 --> 00:04:17,459 Because with a die role where we're rolling a six sided die, of course we're never going to find the 50 00:04:17,459 --> 00:04:20,519 value one and one half or the value two and a half. 51 00:04:20,519 --> 00:04:26,190 We'll never get values in between these six discrete countable values. 52 00:04:26,190 --> 00:04:33,180 So we would never have a horizontal graph that stretches along parallel to this horizontal axis as if 53 00:04:33,180 --> 00:04:38,670 it was defining probability at all of these different values of x along the horizontal axis. 54 00:04:38,670 --> 00:04:44,730 Instead, we have to use this vertical measure to show the probability just at exactly x equals one 55 00:04:44,730 --> 00:04:51,330 and at exactly x equals to exactly x equals three, etc. Whereas up here when we have that continuous 56 00:04:51,330 --> 00:04:58,260 random variable, we need these horizontal pieces of graph to show probability all along the x axis. 57 00:04:58,590 --> 00:04:59,880 Now, with that in mind, let's. 58 00:05:00,030 --> 00:05:05,910 Talk about the probability density function for this continuous random variable x, because in the case 59 00:05:05,910 --> 00:05:11,730 of a continuous uniform distribution, we do have a continuous random variable and every continuous 60 00:05:11,730 --> 00:05:14,520 random variable has a probability density function. 61 00:05:14,730 --> 00:05:20,310 So the probability density function in the case of a continuous uniform distribution is going to be 62 00:05:20,310 --> 00:05:20,910 this function. 63 00:05:20,910 --> 00:05:22,920 Here it's always this function. 64 00:05:22,920 --> 00:05:23,670 Exactly. 65 00:05:23,670 --> 00:05:31,380 So the probability density function F of x is going to be given by one divided by B minus A when the 66 00:05:31,380 --> 00:05:38,310 continuous random variable takes on values between A and B, otherwise the function is going to be zero 67 00:05:38,310 --> 00:05:39,120 everywhere else. 68 00:05:39,120 --> 00:05:49,410 So that means zero when X is less than a so over here to the left of A or when x is greater than B, 69 00:05:49,410 --> 00:05:55,350 So over here to the right of B, so we're zero to the left of A, we're zero to the right of B, but 70 00:05:55,350 --> 00:06:01,170 between A and B, the probability density function is given by this probability here one over B minus 71 00:06:01,170 --> 00:06:05,490 A, and we can figure out why we have one over B minus A. 72 00:06:05,520 --> 00:06:11,310 If we think about what we know about probability density functions, remember that under the curve of 73 00:06:11,310 --> 00:06:17,910 a probability density function, the total area everywhere under the curve has to be equal to one. 74 00:06:17,910 --> 00:06:23,430 That was one of the requirements for something to qualify as a probability density function, which 75 00:06:23,430 --> 00:06:27,270 means that the area all this area under here. 76 00:06:27,970 --> 00:06:31,280 Has to sum up geometrically to one. 77 00:06:31,300 --> 00:06:35,240 And of course, we can see that this is a rectangular shape here. 78 00:06:35,260 --> 00:06:38,800 So the area of this rectangle has to be equal to one. 79 00:06:39,160 --> 00:06:45,490 Well, we know from geometry that the area of a rectangle is equal to the length times the height. 80 00:06:45,520 --> 00:06:50,710 Well, if we look at the length of the rectangle in this case, let's call that the base here of the 81 00:06:50,710 --> 00:06:57,530 rectangle between A and B, We know that this width, this length is going to be B minus A. 82 00:06:57,550 --> 00:07:01,070 Imagine that A is three and that B is five. 83 00:07:01,090 --> 00:07:06,310 We can see right away that the length of this base here is two, because to get from 3 to 5, that's 84 00:07:06,310 --> 00:07:08,040 a width there of two. 85 00:07:08,050 --> 00:07:11,850 So to get the width of two we can see that we just have to take five minus three. 86 00:07:11,860 --> 00:07:16,510 Well, even if we don't know the values of A and B, we can just say that that width, the width of 87 00:07:16,510 --> 00:07:18,310 the base there is B minus A. 88 00:07:18,580 --> 00:07:21,460 So we can say here that this length is. 89 00:07:22,120 --> 00:07:28,330 B minus A, and we don't know the height yet, but we know that the area of this rectangle has to be 90 00:07:28,330 --> 00:07:34,870 one in order for us to have a probability density function, which means that when we divide both sides 91 00:07:34,870 --> 00:07:41,080 of this equation by B minus A, we're just left with height over here on the right hand side and we 92 00:07:41,080 --> 00:07:45,700 get one over B minus A is equal to the height. 93 00:07:45,970 --> 00:07:51,490 And so what that says is that in order for this to be a probability density function, the height of 94 00:07:51,490 --> 00:08:00,130 this rectangle here has to be one over B minus A, and that's why we see the height over here along 95 00:08:00,130 --> 00:08:06,430 the vertical axis given as one over B minus A, and that's why our probability density function gives 96 00:08:06,430 --> 00:08:09,290 this value here as one over B minus A. 97 00:08:09,340 --> 00:08:15,040 So if we have a continuous uniform distribution with this kind of probability density function, where 98 00:08:15,040 --> 00:08:18,640 the height here of this distribution is one over B minus A. 99 00:08:18,670 --> 00:08:25,090 Keep in mind that this is such a simple distribution that will often see this continuous random variable 100 00:08:25,090 --> 00:08:31,120 x written this way with a capital U as the uniform distribution. 101 00:08:32,179 --> 00:08:39,200 Over A to B this interval here A to B, And so then if we have values of A and B, let's say that in 102 00:08:39,200 --> 00:08:47,000 this case A is three and B is five, then we could describe the continuous random variable capital X 103 00:08:47,000 --> 00:08:52,790 as given by the continuous uniform distribution on the interval 3 to 5. 104 00:08:52,790 --> 00:08:59,720 Once we know that we have a continuous uniform distribution for our continuous random variable, then 105 00:08:59,720 --> 00:09:05,870 of course, because this is a standard distribution that we're used to seeing all the time in statistics, 106 00:09:05,870 --> 00:09:11,690 we can unlock mean variance and standard deviation formulas for this kind of distribution. 107 00:09:11,810 --> 00:09:13,850 Which are these values here? 108 00:09:13,850 --> 00:09:19,310 The mean of a continuous uniform distribution is always given by B plus A over two. 109 00:09:19,310 --> 00:09:26,960 So in this case, if A is three and B is five, the mean would be given by five, plus three is eight, 110 00:09:26,960 --> 00:09:28,940 eight divided by two is four. 111 00:09:28,940 --> 00:09:36,800 So the mean the expected value is four, which makes sense because the entire distribution is located 112 00:09:36,830 --> 00:09:44,300 here on this interval A to B and because of the uniform height of the distribution, we expect that 113 00:09:44,300 --> 00:09:50,720 mean to occur right in the center of this rectangle, which in our case is halfway between three and 114 00:09:50,720 --> 00:09:57,760 five or write it for the variance is always given by B minus a quantity squared divided by 12. 115 00:09:57,770 --> 00:10:04,070 This is a fact that we know about all continuous uniform distributions, and so we can use that formula 116 00:10:04,070 --> 00:10:05,630 to quickly find variance. 117 00:10:05,630 --> 00:10:12,650 In our case, that's five minus three is two, two squared is four, four divided by 12 is one over 118 00:10:12,650 --> 00:10:13,250 three. 119 00:10:13,250 --> 00:10:18,170 So the variance is one third or approximately 0.33. 120 00:10:18,170 --> 00:10:22,460 And then of course the standard deviation is always the square root of variance. 121 00:10:22,460 --> 00:10:27,230 So in our case, that's approximately equal to the square root of zero. 122 00:10:28,450 --> 00:10:29,530 .33. 123 00:10:29,530 --> 00:10:34,630 And just because of the fact that we know that this is a continuous uniform distribution for a continuous 124 00:10:34,630 --> 00:10:39,430 random variable, and these are formulas for the mean variance and standard deviation that have already 125 00:10:39,430 --> 00:10:41,860 been established for this kind of distribution. 126 00:10:41,860 --> 00:10:47,200 Once we know we have that kind of distribution, we unlock these and we can quickly get to mean variance 127 00:10:47,200 --> 00:10:50,560 in standard deviation for this continuous random variable. 128 00:10:50,680 --> 00:10:54,060 So that's the idea of a continuous uniform distribution. 129 00:10:54,070 --> 00:10:58,780 We just wanted to make sure that we understood this so that we could contrast it with what we learned 130 00:10:58,780 --> 00:11:05,230 earlier about the discrete uniform distribution when we were talking about discrete random variables 131 00:11:05,230 --> 00:11:10,750 versus this continuous uniform distribution when we're talking about continuous random variables.