1 00:00:00,090 --> 00:00:05,100 Now that we understand the idea of a probability math function, we want to talk about the special case 2 00:00:05,100 --> 00:00:10,920 of the uniform probability distribution, which is where all of the values that the discrete random 3 00:00:10,920 --> 00:00:16,520 variable can take on have equally likely outcomes or an equal probability of occurring. 4 00:00:16,530 --> 00:00:24,060 So we looked at before this general form of a probability mass function where we said if we have a discrete 5 00:00:24,060 --> 00:00:30,510 random variable capital x, the form of our probability mass function is written this way where we say 6 00:00:30,510 --> 00:00:37,260 that the probability that the discrete random variable takes on the specific value x one is given here. 7 00:00:37,260 --> 00:00:42,360 And then we say that this probability occurs when the discrete random variable takes on the value x 8 00:00:42,360 --> 00:00:42,780 one. 9 00:00:42,780 --> 00:00:49,200 And then on the second line here, we give the next possible value that X can have and its associated 10 00:00:49,200 --> 00:00:57,150 probability, and we keep adding more and more lines until we run out of values x one, x two, x three, 11 00:00:57,150 --> 00:01:01,790 etc. that the random variable capital x can possibly have. 12 00:01:01,800 --> 00:01:04,019 So that's our general form. 13 00:01:04,019 --> 00:01:09,840 When we have a discrete uniform distribution, this is what happens to that probability mass function. 14 00:01:10,050 --> 00:01:13,020 All those probabilities are constant. 15 00:01:13,020 --> 00:01:19,860 And so if we say that that constant probability is equal to C, so C can take on any value between zero 16 00:01:19,860 --> 00:01:25,980 and one, but that constant probability is C for all of the possible values of x x sub. 17 00:01:25,980 --> 00:01:33,840 I like these x1x2x3x4 values that we can see for this discrete random variable capital x, and then 18 00:01:33,840 --> 00:01:37,140 the probability is zero for any other values. 19 00:01:37,140 --> 00:01:43,050 In that context, one of the simplest examples of a discrete uniform distribution is the probability 20 00:01:43,050 --> 00:01:43,680 distribution. 21 00:01:43,680 --> 00:01:47,370 We see four dice rolls when we roll a six sided die. 22 00:01:47,400 --> 00:01:52,920 So if we were to build that probability mass function, here's what we need to think about. 23 00:01:52,920 --> 00:01:59,640 If we roll a six sided die one time, we know, of course, that the values we can get back are one, 24 00:01:59,640 --> 00:02:01,710 two, three, four, five or six. 25 00:02:01,740 --> 00:02:09,270 So we could go ahead and list those possible values one, two, three, four, five, six. 26 00:02:09,270 --> 00:02:12,780 And then we need to find the probability that one of these values occurs. 27 00:02:12,780 --> 00:02:16,800 Well, each of these values has a one in six chance of occurring. 28 00:02:16,800 --> 00:02:21,900 We have a one in six chance of rolling a1a1 in six chance of rolling a to a one in, six chance of rolling 29 00:02:21,900 --> 00:02:24,390 a three, etc. all the way up to six. 30 00:02:24,390 --> 00:02:27,420 We have six equally likely outcomes. 31 00:02:27,420 --> 00:02:31,650 And so for any particular outcome, the probability is one in six. 32 00:02:31,650 --> 00:02:35,880 So we could write that as one in six or we could convert it to a decimal. 33 00:02:35,910 --> 00:02:40,320 The decimal value is approximately 0.167. 34 00:02:40,320 --> 00:02:46,380 So in our probability math function here, we're saying that the probability of getting a one is 0.167. 35 00:02:46,380 --> 00:02:49,380 The probability of getting a two is 0.167. 36 00:02:49,380 --> 00:02:55,680 The probability of getting a three is 0.1, six seven, etc. And these are all of the possible values, 37 00:02:55,680 --> 00:03:02,820 all of the x values for the discrete random variable x, so the probability of anything else is zero. 38 00:03:02,820 --> 00:03:07,050 And so we could say otherwise the probability is zero. 39 00:03:07,230 --> 00:03:12,510 And this makes sense as a probability mass function because this is a discrete random variable. 40 00:03:12,510 --> 00:03:15,630 We have a countable set of discrete values. 41 00:03:15,630 --> 00:03:20,490 There are six values the variable can take on and they are discrete, they're countable. 42 00:03:20,490 --> 00:03:22,560 There are no in-between values. 43 00:03:22,560 --> 00:03:27,300 We can't measure in between two and three to get a dice roll of 2.5. 44 00:03:27,300 --> 00:03:28,440 These are the only values. 45 00:03:28,440 --> 00:03:33,990 So we have a discrete random variable so we can say that the random variable x is. 46 00:03:34,660 --> 00:03:35,590 Discrete. 47 00:03:35,590 --> 00:03:40,210 So that's our first criteria for the probability mass function. 48 00:03:40,240 --> 00:03:46,240 The second criteria is that the probability of each of those values is between zero and one. 49 00:03:46,240 --> 00:03:49,950 And therefore also by definition, that none of those values are negative. 50 00:03:49,960 --> 00:03:51,490 And we can see that that's the case. 51 00:03:51,490 --> 00:03:56,280 This 0.167 value is between zero and one, and it's not negative. 52 00:03:56,290 --> 00:04:01,710 So this probability associated with each of these outcomes meets that criteria. 53 00:04:01,720 --> 00:04:09,640 We can say that all of the p of x sub P values are between zero and one. 54 00:04:09,640 --> 00:04:16,060 So we meet that criteria and then these values have to sum to a total of one. 55 00:04:16,060 --> 00:04:24,670 And we have this 0.167 probability six times and 0.167 APR times six is equal to one. 56 00:04:24,670 --> 00:04:32,710 So the probability of all of the values of X is one, and we meet that criteria as well. 57 00:04:32,800 --> 00:04:38,500 Think about this another way to right this probability mass function for the discrete uniform distribution 58 00:04:38,500 --> 00:04:48,220 is if we broke it out like this, we could say 0.16700.167 for x equals one. 59 00:04:48,220 --> 00:04:57,640 We could say 0.167 for x equals to 0.167 for x equals three. 60 00:04:57,640 --> 00:05:02,050 And we could keep going, putting each value on its own line. 61 00:05:02,050 --> 00:05:04,540 We've just summarized the probability mass function. 62 00:05:04,540 --> 00:05:07,600 We've consolidated it by putting everything on one line. 63 00:05:07,600 --> 00:05:10,150 But this says the same thing that this does. 64 00:05:10,150 --> 00:05:13,900 It's just consolidated into one line instead of six. 65 00:05:13,900 --> 00:05:16,180 But we have our probability mass function. 66 00:05:16,180 --> 00:05:19,990 We have it described algebraically this way now we can go ahead and graph it. 67 00:05:19,990 --> 00:05:22,240 And that looks like this. 68 00:05:22,240 --> 00:05:29,920 We have along the horizontal axis here all of the possible equally likely outcomes of our die role. 69 00:05:29,920 --> 00:05:32,320 We can roll anywhere from a one to a six. 70 00:05:32,320 --> 00:05:37,810 And then the probability that our discrete random variable takes on any of those specific values of 71 00:05:37,810 --> 00:05:39,040 x is p of x. 72 00:05:39,040 --> 00:05:42,820 Here, our probability for each of those values is 0.167. 73 00:05:42,820 --> 00:05:49,960 So if we sketch in this graph, we see here a uniform probability across each of these equally likely 74 00:05:49,960 --> 00:05:50,590 outcomes. 75 00:05:50,590 --> 00:05:57,760 And that is why we call it a uniform distribution, because the probability distribution is uniform, 76 00:05:57,760 --> 00:06:01,090 it stays the same no matter which value we choose. 77 00:06:01,090 --> 00:06:08,950 For X, we can see that the value is level and constant everywhere along this horizontal axis for all 78 00:06:08,950 --> 00:06:12,790 of these different values of X, and that value is C. 79 00:06:12,820 --> 00:06:21,460 If we're talking about this form here where that probability P of X stays level or uniform at this constant 80 00:06:21,490 --> 00:06:22,030 C. 81 00:06:22,030 --> 00:06:26,320 So that's the simple idea behind a uniform distribution. 82 00:06:26,320 --> 00:06:30,850 But as this die roll example indicates, we see it all the time in the real world. 83 00:06:30,850 --> 00:06:36,850 So it's important for us to understand that when we have uniform probability across the full set of 84 00:06:36,850 --> 00:06:43,930 outcomes, the result is this discrete uniform distribution, the uniform distribution of the discrete 85 00:06:43,930 --> 00:06:45,130 random variable.