1 00:00:00,120 --> 00:00:06,510 Cumulative distribution functions model, the probability that some random variable will take on a value 2 00:00:06,510 --> 00:00:09,260 less than some specific value that we set. 3 00:00:09,270 --> 00:00:11,360 So we model them this way. 4 00:00:11,370 --> 00:00:12,990 This is what the equation looks like. 5 00:00:12,990 --> 00:00:19,200 We distinguish between probability mass functions and probability density functions and cumulative distribution 6 00:00:19,200 --> 00:00:24,840 functions by using this capital F, So with probability mass and probability density functions, we 7 00:00:24,840 --> 00:00:26,550 used a lowercase F. 8 00:00:26,550 --> 00:00:32,549 If you see a capital F, it means that we have transitioned to a cumulative distribution function instead. 9 00:00:32,549 --> 00:00:39,300 And what this says here is that the cumulative distribution function is equal to this is the probability 10 00:00:39,300 --> 00:00:45,900 that our discrete random variable capital X takes on some value less than or equal to this value of 11 00:00:45,900 --> 00:00:47,310 x that we set. 12 00:00:47,340 --> 00:00:54,630 Now, we actually understand this better if we look at the graphs of the distributions for both discrete 13 00:00:54,630 --> 00:00:56,670 and continuous random variables. 14 00:00:56,670 --> 00:00:59,370 So let's start with a discrete random variable. 15 00:00:59,370 --> 00:01:02,040 So we're talking about discrete random variables here. 16 00:01:02,370 --> 00:01:07,140 We have already looked at the probability mass function for a discrete random variable. 17 00:01:07,140 --> 00:01:10,440 This is the one we've been using throughout these lessons. 18 00:01:10,440 --> 00:01:15,750 It's the probability mass function associated with rolling one six sided die one time. 19 00:01:15,750 --> 00:01:21,240 And it shows us that a probability that we get a one, two, three, four, five or six when we roll 20 00:01:21,240 --> 00:01:23,610 the die is always equivalent. 21 00:01:23,610 --> 00:01:26,310 That's why we see that all of these bars have equal heights. 22 00:01:26,310 --> 00:01:28,590 The probability is always one over six. 23 00:01:28,590 --> 00:01:33,540 The probability of rolling a one is one over six, The probability of rolling a two is one over six, 24 00:01:33,540 --> 00:01:40,350 etc. So this is the probability distribution showing the probability mass function for the discrete 25 00:01:40,350 --> 00:01:41,340 random variable. 26 00:01:41,340 --> 00:01:47,550 This graph on the right, on the other hand, is the cumulative distribution function that's associated 27 00:01:47,550 --> 00:01:50,100 with this same probability mass function. 28 00:01:50,100 --> 00:01:56,610 So if we were to transfer this information from the probability distribution over here to this cumulative 29 00:01:56,610 --> 00:02:00,180 distribution, this is how one graph translates to the other. 30 00:02:00,180 --> 00:02:05,700 In other words, this is the same probability here, the probability associated with rolling a six sided 31 00:02:05,700 --> 00:02:06,870 die one time. 32 00:02:06,870 --> 00:02:09,150 We're just showing the cumulative distribution. 33 00:02:09,150 --> 00:02:15,270 And so what we're saying here, when we look at the graph or what this equation over here tells us is 34 00:02:15,270 --> 00:02:19,020 that we can define this kind of cumulative probability. 35 00:02:19,020 --> 00:02:25,500 So instead of just saying what is the probability that we roll a one or roll a four or roll a six, 36 00:02:25,500 --> 00:02:31,560 this is saying what is the probability that we roll a number less than or equal to two, less than or 37 00:02:31,560 --> 00:02:34,410 equal to four or less than or equal to six, for example. 38 00:02:34,410 --> 00:02:39,630 So that word cumulative is appropriate because the probability accumulates. 39 00:02:39,630 --> 00:02:47,400 And that idea of accumulation means that the cumulative distribution is always going to be increasing. 40 00:02:47,400 --> 00:02:53,610 And in fact, because we know that probability is always defined between zero and one over here on the 41 00:02:53,610 --> 00:03:00,120 left edge of the graph of our cumulative distribution function, the probability that we see is zero. 42 00:03:00,150 --> 00:03:05,340 The value of the graph along the vertical axis here is at zero, and that probability is always going 43 00:03:05,340 --> 00:03:10,920 to increase and increase and increase as we move to the right until eventually it gets to one. 44 00:03:10,920 --> 00:03:16,230 We see here we have six over six or one, the probability tops out at one. 45 00:03:16,530 --> 00:03:23,220 And so we say that the cumulative distribution has these properties, that the limit as X gets close 46 00:03:23,220 --> 00:03:29,550 to negative infinity is zero, and that the limit as x gets close to positive infinity is one. 47 00:03:29,730 --> 00:03:34,890 Now this limit concept comes from calculus, and that's really beyond the scope of this course. 48 00:03:34,890 --> 00:03:40,260 But technically all this is really saying is that we're at a probability of zero over here on the left. 49 00:03:40,260 --> 00:03:42,060 That's where this zero comes from. 50 00:03:42,180 --> 00:03:44,460 And one over here on the right. 51 00:03:44,460 --> 00:03:46,190 That's where this one comes from. 52 00:03:46,200 --> 00:03:51,450 In other words, our cumulative distribution, the graph of our cumulative distribution is always going 53 00:03:51,450 --> 00:03:58,320 to start at one and always increase and always end at positive one, which makes sense because again, 54 00:03:58,320 --> 00:04:01,890 we are accumulating probability as we move to the right. 55 00:04:01,890 --> 00:04:08,130 If we're rolling a six sided die one time, the probability that we roll a value less than one is of 56 00:04:08,130 --> 00:04:12,510 course going to be zero, because the lowest value we can roll is a one itself. 57 00:04:12,510 --> 00:04:17,670 That's why we see this zero value along the horizontal axis for these values. 58 00:04:17,670 --> 00:04:23,730 Less than one these empty circles here indicate that this value is not a part of the graph, whereas 59 00:04:23,730 --> 00:04:27,990 these solid filled in circles indicate that this is a part of the graph. 60 00:04:27,990 --> 00:04:33,870 So what this tells us is that the probability that we roll a value less than or equal to one, we come 61 00:04:33,870 --> 00:04:40,410 over to one along the horizontal axis and then we go up and find the solid part of the graph, which 62 00:04:40,410 --> 00:04:43,080 is not this empty part here, but the solid part here. 63 00:04:43,080 --> 00:04:49,440 We come over to the vertical axis and we see that this is one over six. 64 00:04:49,440 --> 00:04:55,260 So the probability that we roll a value less than or equal to one is one sixth, the probability that 65 00:04:55,260 --> 00:04:59,870 we roll a value, let's say less than or equal to three, we come over to three. 66 00:05:00,390 --> 00:05:02,910 We go up until we find a solid part of the graph. 67 00:05:02,910 --> 00:05:04,710 So we skip past this empty circle. 68 00:05:04,740 --> 00:05:06,210 We come up here. 69 00:05:06,240 --> 00:05:09,710 Then we go over along to the vertical axis. 70 00:05:09,720 --> 00:05:11,940 This is at three over six. 71 00:05:12,090 --> 00:05:17,310 So the probability that we roll a value less than or equal to three is three over six. 72 00:05:17,310 --> 00:05:22,590 And that makes sense because less than or equal to three means we're rolling a one, a two or three, 73 00:05:22,590 --> 00:05:28,440 which represents half of the total possibilities and three over six is equal to one half. 74 00:05:28,440 --> 00:05:35,640 So we can see how those dice rolls are accumulating and eventually we get to x equals six here where 75 00:05:35,640 --> 00:05:42,060 we can say we come up to the solid part of the graph and then we come over to the vertical axis and 76 00:05:42,060 --> 00:05:46,590 we can see the probability six over six or one there, this is one. 77 00:05:46,830 --> 00:05:53,010 So the probability that we roll a value less than or equal to six is of course, one. 78 00:05:53,010 --> 00:05:56,640 So that's what the cumulative distribution function is looking like. 79 00:05:56,640 --> 00:05:58,860 For a discrete random variable. 80 00:05:58,860 --> 00:06:01,650 We are adding up all of those probabilities as we go. 81 00:06:01,680 --> 00:06:07,440 It would be like taking this graph of the probability mass function and adding each of these probabilities 82 00:06:07,440 --> 00:06:08,190 as we go. 83 00:06:08,190 --> 00:06:12,830 So the probability that we roll a one or less is one sixth. 84 00:06:12,900 --> 00:06:18,360 Then when we add another one six to that, we can say that the probability of rolling a two or less 85 00:06:18,390 --> 00:06:20,760 is two over six or one third. 86 00:06:20,790 --> 00:06:28,080 The probability of rolling a three or less is one sixth plus one sixth plus one sixth or three over 87 00:06:28,080 --> 00:06:29,430 six or one half. 88 00:06:29,430 --> 00:06:34,440 To give us yet another view on this, let's look at continuous random variables. 89 00:06:34,440 --> 00:06:36,420 So it's the same idea. 90 00:06:36,420 --> 00:06:41,700 We can use this cumulative distribution function for both discrete and continuous random variables. 91 00:06:41,700 --> 00:06:47,460 When we have a continuous random variable, the graph of our probability density function might look 92 00:06:47,460 --> 00:06:53,730 something like this, and if we transfer that over into a cumulative distribution function, we see 93 00:06:53,730 --> 00:06:59,790 how this probability accumulates starting down at zero and continuing to accumulate probability all 94 00:06:59,790 --> 00:07:07,650 the way up until we get to one, at which point it levels off and never rises above this value of one. 95 00:07:07,650 --> 00:07:14,520 And we can imagine here that this sort of mean value in the probability density function is represented 96 00:07:14,520 --> 00:07:17,220 here in the cumulative distribution function. 97 00:07:17,220 --> 00:07:19,800 So we have those visuals. 98 00:07:19,800 --> 00:07:26,610 If we wanted to plug into the actual function here, let's say we wanted to model the probability for 99 00:07:26,610 --> 00:07:29,670 this discrete random variable representing the DI role. 100 00:07:29,670 --> 00:07:34,380 We wanted to model the probability of rolling something less than or equal to four. 101 00:07:34,410 --> 00:07:40,530 Then our cumulative distribution function here would be capital F and then capital X, and then we would 102 00:07:40,530 --> 00:07:48,570 say of four lowercase x Equal to four is the probability that the discrete random variable x takes on 103 00:07:48,570 --> 00:07:51,330 a value less than or equal to four. 104 00:07:51,450 --> 00:07:57,030 If we wanted to say here for the continuous random variable, let's say that maybe this right here is 105 00:07:57,030 --> 00:08:03,120 x equals two and maybe roughly the same here as something like this x equals two, Then we could do 106 00:08:03,120 --> 00:08:08,640 the exact same thing for the continuous random variable and say that the cumulative distribution function 107 00:08:08,640 --> 00:08:16,320 we have capital F, capital X of two is equal to the probability that the continuous random variable 108 00:08:16,320 --> 00:08:20,070 represented by Capital X is less than or equal to two. 109 00:08:20,100 --> 00:08:22,530 We would get that cumulative probability. 110 00:08:22,530 --> 00:08:29,130 The probability that this continuous random variable is taking on any value less than two or the exact 111 00:08:29,130 --> 00:08:29,880 value of two. 112 00:08:29,910 --> 00:08:32,549 So anything up to and including two. 113 00:08:32,549 --> 00:08:40,710 And we could find that probability on the graph by coming up from two here to the corresponding point 114 00:08:40,710 --> 00:08:43,770 on the graph and then coming over here to the vertical axis. 115 00:08:43,770 --> 00:08:51,870 And this value along the vertical axis, let's say that's about 0.85 based on where we put in this one 116 00:08:51,870 --> 00:08:57,630 here, I'm just looking at the scale between zero and one and estimating that that's about it, 0.85. 117 00:08:57,630 --> 00:09:03,090 And so maybe if this is the shape of the cumulative distribution function, then the probability that 118 00:09:03,090 --> 00:09:09,060 the continuous random variable takes on a value less than or equal to two is approximately 0.85 or 85. 119 00:09:09,600 --> 00:09:16,080 So again, instead of just modeling the probability of the discrete random variable, taking on a specific 120 00:09:16,080 --> 00:09:21,960 value, taking on the specific value x equals three or for the continuous random variable in the probability 121 00:09:21,960 --> 00:09:28,140 density function, the probability that we take on a value between, let's say one and two. 122 00:09:28,170 --> 00:09:30,330 So take on some value between one and two. 123 00:09:30,330 --> 00:09:35,370 That's what we did before with probability mass functions and probability density functions for discrete 124 00:09:35,370 --> 00:09:37,590 and continuous random variables respectively. 125 00:09:37,590 --> 00:09:42,840 Now, when we switch to cumulative distribution functions, we're looking at the probability that the 126 00:09:42,840 --> 00:09:50,640 random variable takes on some value up to some specified value up to some specified value instead of 127 00:09:50,640 --> 00:09:55,740 the probability of an exact value or the probability of some value in an interval. 128 00:09:55,740 --> 00:09:59,240 And then the only the thing we want to say about this is that we can. 129 00:09:59,600 --> 00:10:05,900 Use a cumulative distribution to find the probability that we do get a value in a particular interval. 130 00:10:05,900 --> 00:10:07,850 And we use this formula here. 131 00:10:07,850 --> 00:10:13,730 So if we have a cumulative distribution function, whether for a discrete random variable or a continuous 132 00:10:13,730 --> 00:10:19,280 random variable, we can use the values along the cumulative distribution function to find the probability 133 00:10:19,280 --> 00:10:22,340 that the random variable takes on a value in that interval. 134 00:10:22,340 --> 00:10:27,020 So, for instance, going back to this discrete random variable, because we have concrete values here, 135 00:10:27,020 --> 00:10:35,330 it's easy to see if we want to know the probability that when we roll the die, we get a value, let's 136 00:10:35,330 --> 00:10:39,290 say between two and four. 137 00:10:39,290 --> 00:10:43,460 And notice this here notation less than and less than or equal to. 138 00:10:43,490 --> 00:10:50,030 So what we're saying is that we're looking for the probability that X is between two and four, not 139 00:10:50,030 --> 00:10:52,400 including two, but including four. 140 00:10:52,400 --> 00:10:58,910 So if we have an interval here, two, three, four, we're not including two, but we are including 141 00:10:58,910 --> 00:10:59,270 four. 142 00:10:59,270 --> 00:11:03,590 So the only values that would actually fall within this interval are the values three and four. 143 00:11:03,590 --> 00:11:10,490 So we set that up and then we look at our cumulative distribution indicated by this capital F and we 144 00:11:10,490 --> 00:11:15,650 find the value of that cumulative distribution function at B, So B is four. 145 00:11:15,650 --> 00:11:18,380 We come here to four and we see that up at four. 146 00:11:18,380 --> 00:11:21,590 Here we're at four over six, which is two thirds. 147 00:11:21,590 --> 00:11:28,160 So we get two thirds and then we subtract the value of the cumulative distribution function evaluated 148 00:11:28,160 --> 00:11:30,950 at a well, A is two. 149 00:11:30,950 --> 00:11:36,620 So we come up here to two, we come up to the solid circle, come over to the vertical axis, and we 150 00:11:36,620 --> 00:11:39,410 see they're two over six or one third. 151 00:11:39,410 --> 00:11:45,590 So we subtract one third and the result then two over three minus one over three is one. 152 00:11:46,290 --> 00:11:47,010 Third. 153 00:11:47,010 --> 00:11:52,470 So this tells us that the probability that we get a value between two and four, not including two, 154 00:11:52,470 --> 00:11:54,900 but including four is one third. 155 00:11:54,930 --> 00:12:01,950 And of course, that makes sense because we said here that this expression showing this interval implies 156 00:12:01,950 --> 00:12:03,750 getting either a three or a four. 157 00:12:03,780 --> 00:12:06,870 Those are the only two values that fall within this interval. 158 00:12:06,870 --> 00:12:12,660 And of course, out of six possibilities when we roll the dice by getting anywhere from a one to a six, 159 00:12:12,660 --> 00:12:14,310 there are six possibilities. 160 00:12:14,310 --> 00:12:19,650 There are two possibilities of those six that meet this criteria that fall within this interval. 161 00:12:19,740 --> 00:12:24,660 So out of the six total possibilities, there are two that fall within the interval. 162 00:12:24,660 --> 00:12:30,090 And of course that simplifies to one third, one third of our possibilities. 163 00:12:30,090 --> 00:12:31,890 Meet this interval, meet this criteria. 164 00:12:31,890 --> 00:12:35,700 And so the probability that we fall within this interval is one third. 165 00:12:35,700 --> 00:12:41,730 So we can also use our cumulative distribution, whether it's for a discrete or continuous random variable 166 00:12:41,730 --> 00:12:48,030 to calculate the probability that the random variable will take on a value in this particular interval.