1 00:00:00,060 --> 00:00:06,190 We said before that a binomial random variable was a special type of discrete random variable. 2 00:00:06,210 --> 00:00:12,660 And what we want to say now is that this Bernoulli random variable is a special kind of binomial random 3 00:00:12,660 --> 00:00:13,290 variable. 4 00:00:13,290 --> 00:00:19,590 And the difference is just that with the Bernoulli random variable, we're modeling exactly one trial, 5 00:00:19,590 --> 00:00:23,400 whereas with the binomial random variable, we were modeling many trials. 6 00:00:23,400 --> 00:00:29,610 So before if we used a binomial random variable to model the probability of flipping a certain number 7 00:00:29,610 --> 00:00:35,520 of heads on a larger number of coin flips, we might have been flipping the coin five times or ten times 8 00:00:35,520 --> 00:00:36,420 or 1000 times. 9 00:00:36,420 --> 00:00:38,250 We had many trials. 10 00:00:38,250 --> 00:00:43,170 When we use a Bernoulli random variable, we drill that down to just one trial. 11 00:00:43,170 --> 00:00:49,890 So all we're doing here is flipping a coin one time, and in that scenario we can use a Bernoulli random 12 00:00:49,890 --> 00:00:52,050 variable to model probability. 13 00:00:52,050 --> 00:00:57,780 We can also, of course build our Bernoulli distribution, which will be a visual representation of 14 00:00:57,780 --> 00:01:01,020 the probability associated with that Bernoulli random variable. 15 00:01:01,050 --> 00:01:09,510 The way that we accomplish using this Bernoulli random variable is by defining success as a one and 16 00:01:09,510 --> 00:01:11,100 failure as a zero. 17 00:01:11,100 --> 00:01:18,360 In other words, we assign numeric mathematical values to these concepts of success and failure. 18 00:01:18,360 --> 00:01:23,760 So if we're flipping a coin one time and we're looking at the probability that we flip heads, we would 19 00:01:23,760 --> 00:01:29,490 define flipping heads as a success and therefore as a one and everything else, in this case, flipping 20 00:01:29,520 --> 00:01:31,890 tails as a failure or zero. 21 00:01:31,890 --> 00:01:39,180 If we're rolling a six sided die one time we have one trial, we could define success as rolling a two. 22 00:01:39,180 --> 00:01:44,340 So rolling a two, we would sort of assign this value of one rolling anything else. 23 00:01:44,340 --> 00:01:49,720 So a one, three, four, five or six would be assigned a failure or a zero value. 24 00:01:49,740 --> 00:01:54,720 It's really this idea of distilling down success and failure to these values. 25 00:01:54,720 --> 00:01:59,490 One and zero that makes this Bernoulli random variable unique and useful. 26 00:01:59,490 --> 00:02:06,570 So if we stick with our example from earlier about the rideshare company where we defined success as 27 00:02:06,570 --> 00:02:14,250 a driver arriving within 10 minutes of the moment when the passenger requests the ride and failure as 28 00:02:14,250 --> 00:02:20,340 a driver taking longer than 10 minutes to arrive After the ride request, we said that the probability 29 00:02:20,340 --> 00:02:28,590 of success was 75%, which means then we know that the Bernoulli distribution looks like this. 30 00:02:28,590 --> 00:02:30,030 We know this right away. 31 00:02:30,030 --> 00:02:35,910 This is just like the binomial distribution in the sense that we have the set of discrete countable 32 00:02:35,910 --> 00:02:38,430 outcomes along the horizontal axis. 33 00:02:38,430 --> 00:02:40,980 In this case, we're using this Bernoulli random variable. 34 00:02:40,980 --> 00:02:43,440 So we assign those values zero and one. 35 00:02:43,440 --> 00:02:49,320 So with the Bernoulli distribution, we will only ever see zero and one along the horizontal axis. 36 00:02:49,320 --> 00:02:54,000 We will always see both zero and one and never any other values. 37 00:02:54,000 --> 00:02:57,150 And then along the vertical axis, we model probability. 38 00:02:57,150 --> 00:03:03,390 So using that rideshare example where we said that the probability of success was 75%, then here at 39 00:03:03,390 --> 00:03:11,040 one we have a 75% probability, and since zero is the probability of failure, just like with the binomial 40 00:03:11,040 --> 00:03:17,730 random variable, if we define the probability of success as P, then the probability of failure has 41 00:03:17,730 --> 00:03:19,140 to be one minus P. 42 00:03:19,170 --> 00:03:24,510 Sometimes we write that as Q, but in this case success is 75%. 43 00:03:24,510 --> 00:03:28,350 So one -75% is 25% or 0.25. 44 00:03:28,350 --> 00:03:34,920 And so we can go ahead and graph in here 0.25 for this probability of failure or the probability of 45 00:03:34,920 --> 00:03:36,330 this idea of zero. 46 00:03:36,330 --> 00:03:42,390 So in this Bernoulli distribution, of course, just like other distributions, these two values here, 47 00:03:42,390 --> 00:03:51,420 the 0.25 value that we see here and the 0.75 value that we see here, these values are always going 48 00:03:51,420 --> 00:03:54,600 to sum to one just like our other distribution. 49 00:03:54,600 --> 00:04:01,020 We're just always going to have only these two values represented zero and one as failure and success 50 00:04:01,020 --> 00:04:01,860 respectively. 51 00:04:01,860 --> 00:04:08,760 Now, once we know that we're dealing with a Bernoulli distribution for a Bernoulli random variable, 52 00:04:08,760 --> 00:04:14,490 just like all of our other distributions that unlocks for us formulas that we already have for mean 53 00:04:14,490 --> 00:04:17,790 variance and standard deviation for Bernoulli distribution. 54 00:04:17,790 --> 00:04:19,380 Here's what those look like. 55 00:04:19,410 --> 00:04:25,890 These are actually adapted directly from the corresponding formulas for a binomial random variable. 56 00:04:25,920 --> 00:04:32,370 The only difference is that they're incorporating these values of one and zero for success and failure. 57 00:04:32,370 --> 00:04:39,840 You can imagine here with this rideshare example, if we wanted to find the mean for just this one particular 58 00:04:39,840 --> 00:04:47,100 trial where our chance of success is 75% and we have these values zero and one, we would find the mean 59 00:04:47,100 --> 00:04:53,760 or the expected value by taking zero, multiplying it by the 25% chance of getting zero. 60 00:04:53,760 --> 00:04:57,720 And then we would add to that the product of one and 0.75. 61 00:04:57,720 --> 00:04:59,850 So our mean would. 62 00:04:59,980 --> 00:05:02,440 Actually look like 0.25. 63 00:05:02,440 --> 00:05:05,800 So 0.25 times zero. 64 00:05:06,660 --> 00:05:09,060 Plus 0.75. 65 00:05:10,140 --> 00:05:11,360 Times one. 66 00:05:11,370 --> 00:05:13,410 This would be our expected value. 67 00:05:13,410 --> 00:05:20,190 If we have these discrete values zero and one, we have a 25% chance of getting quote unquote zero and 68 00:05:20,190 --> 00:05:22,860 a 75% chance of getting quote unquote, one. 69 00:05:22,860 --> 00:05:27,150 And so this is our weighted mean, our expected value. 70 00:05:27,180 --> 00:05:31,140 Of course, we get zero from this first term and we get 0.75 from the second term. 71 00:05:31,140 --> 00:05:33,090 So we get 0.75. 72 00:05:33,090 --> 00:05:38,490 And what we see is that this value here is equivalent to the chance of success. 73 00:05:38,490 --> 00:05:42,990 We defined earlier 0.75 or the chance of getting a one here, 0.75. 74 00:05:42,990 --> 00:05:49,650 And that's always going to be the case because this zero value here is going to zero out this failure 75 00:05:49,650 --> 00:05:56,910 term, which is why we're always going to be left with a mean equal to p the probability of success. 76 00:05:56,910 --> 00:06:03,150 So for a Bernoulli random variable, the mean will always be equal to the chance of success. 77 00:06:03,150 --> 00:06:11,280 P Compare that to the binomial random variable where the mean was equal to n times P where n was the 78 00:06:11,280 --> 00:06:14,390 number of trials and P was the chance of success. 79 00:06:14,400 --> 00:06:19,170 This makes sense also when we think about adapting it this way, because now with the Bernoulli random 80 00:06:19,170 --> 00:06:25,170 variable, which we're saying is a specific kind of binomial random variable, we only have one trial 81 00:06:25,170 --> 00:06:32,010 and so n is equal to one, which means we end up with a mean of one times P, which of course is just 82 00:06:32,010 --> 00:06:32,310 P. 83 00:06:32,310 --> 00:06:37,620 And so of course the mean for the Bernoulli random variable is going to be equal to just P, and we 84 00:06:37,620 --> 00:06:42,750 can use similar logic to get the formulas for variance in standard deviation for the Bernoulli random 85 00:06:42,750 --> 00:06:49,440 variable as adaptations of the formulas for variance in standard deviation of the binomial random variable. 86 00:06:49,440 --> 00:06:53,760 So for example, remember to find variance. 87 00:06:53,760 --> 00:06:59,550 When we have a discrete random variable, we take all of the values that our variable can take on. 88 00:06:59,550 --> 00:07:04,050 So in this case, zero and one, and for each one we subtract the mean. 89 00:07:04,050 --> 00:07:08,010 Well, our mean is P, So we would take here our first value of zero. 90 00:07:08,010 --> 00:07:15,360 So we would say zero, we would subtract the mean, which is p, and then we would square that value. 91 00:07:15,360 --> 00:07:22,560 So this is the square difference between this particular x sub I and the mean, and then we multiply 92 00:07:22,560 --> 00:07:26,310 this by the probability of getting a zero. 93 00:07:26,310 --> 00:07:30,330 Well, in this case that's the probability of failure one minus P. 94 00:07:30,330 --> 00:07:37,050 So we multiply that by one minus P, and we do this for every value that X can possibly take on. 95 00:07:37,050 --> 00:07:39,390 So the only other value here is one. 96 00:07:39,390 --> 00:07:48,360 So we add to this here we have one minus the mean, we square that difference and then we multiply this 97 00:07:48,360 --> 00:07:54,600 by the probability that this value occurs, which in this case is 0.75, it's the probability of success, 98 00:07:54,600 --> 00:07:57,540 it's P, so we multiply by P. 99 00:07:57,720 --> 00:08:00,240 So this is the sum that we find for the variance. 100 00:08:00,240 --> 00:08:07,890 And when we simplify this equation here, we get zero minus P is a negative P, negative P quantity 101 00:08:07,890 --> 00:08:10,170 squared is positive P squared. 102 00:08:10,170 --> 00:08:19,080 So we could say P squared times one minus P, and then here we have p times one minus P quantity squared. 103 00:08:19,080 --> 00:08:26,010 Well, when we expand one minus P quantity squared, that's like saying one minus P times one minus 104 00:08:26,040 --> 00:08:28,110 P, This goes back to algebra. 105 00:08:28,110 --> 00:08:31,590 We have to multiply our first terms one and one. 106 00:08:32,010 --> 00:08:33,990 So one times one is one. 107 00:08:34,169 --> 00:08:42,750 Then we multiply our outer terms one and negative P one times negative P is negative P, Then we multiply 108 00:08:42,750 --> 00:08:50,280 our inner terms negative P times one is a negative P, and then our last terms negative P times negative 109 00:08:50,280 --> 00:08:52,050 P is a positive. 110 00:08:52,820 --> 00:08:53,910 P squared. 111 00:08:53,930 --> 00:08:57,440 And when we simplify again, this is all just algebra. 112 00:08:57,440 --> 00:08:58,640 We get P squared times. 113 00:08:58,640 --> 00:09:08,330 One is P squared, P squared times a negative P is minus P cubed, we'll say plus p times one is P. 114 00:09:08,720 --> 00:09:14,960 Here we have minus P and minus P, that's a minus to P. 115 00:09:15,170 --> 00:09:24,980 So when we multiply P by -$0.02, we get -$0.02 squared and then p times P squared is P cubed. 116 00:09:24,980 --> 00:09:29,660 We get our negative P cubed and our positive P cubed to cancel with each other. 117 00:09:29,660 --> 00:09:33,230 And then we're just left with we'll start with this P here. 118 00:09:33,230 --> 00:09:42,260 So P and then we have P squared -$0.02 squared is a minus P squared, or we can write that as the variance 119 00:09:42,290 --> 00:09:47,930 is equal to when we factor out a P here, we get P times one minus P. 120 00:09:47,930 --> 00:09:52,100 In other words, we rewrote P minus P squared as p times one minus P. 121 00:09:52,100 --> 00:09:57,980 And you see here how we build this formula for variance that we already established over here as the 122 00:09:57,980 --> 00:10:00,200 variance of a newly random variable. 123 00:10:00,200 --> 00:10:04,310 Now if you're not super comfortable with algebra, that's okay. 124 00:10:04,340 --> 00:10:11,270 All of these steps in the middle here is just a bunch of algebra to prove to you that we can get from 125 00:10:11,270 --> 00:10:15,440 this first equation here that we built. 126 00:10:15,440 --> 00:10:22,040 From what we already know about variance down to this equation here to prove to you that this variance 127 00:10:22,040 --> 00:10:26,300 formula works, that it's true that it makes sense for a Bernoulli random variable. 128 00:10:26,300 --> 00:10:31,790 But if all we do is walk away from this lesson, understanding that the variance of a Bernoulli random 129 00:10:31,790 --> 00:10:36,680 variable is given by this formula here, then that's totally fine. 130 00:10:36,680 --> 00:10:38,090 This is really the main takeaway. 131 00:10:38,090 --> 00:10:43,670 We want to understand that in order to find variance for a Bernoulli random variable, we use this formula 132 00:10:43,670 --> 00:10:44,210 here. 133 00:10:44,210 --> 00:10:55,100 In our case, that means that variance is equal to P, which is 0.75 times one minus P or Q. 134 00:10:55,130 --> 00:11:05,840 The probability of failure, which is 0.25, that gives us a value for variance of 0.1875. 135 00:11:05,840 --> 00:11:10,130 And then standard deviation is always just the square root of variance. 136 00:11:10,130 --> 00:11:21,770 So the square root of 0.1875, which is approximately equal to 0.4330. 137 00:11:21,770 --> 00:11:27,110 And so we're able to calculate the mean variance in standard deviation of a Bernoulli random variable. 138 00:11:27,110 --> 00:11:33,710 So the takeaway here is that we're interested in drilling down on this idea of a Bernoulli random variable, 139 00:11:33,740 --> 00:11:37,880 because the thinking behind it is interesting what we're really saying here. 140 00:11:37,880 --> 00:11:45,290 Let's take our rideshare example where we're saying that there's always a 75% chance that a driver arrives 141 00:11:45,290 --> 00:11:48,500 within 10 minutes of the ride being requested. 142 00:11:48,500 --> 00:11:55,160 We're saying that if we distill that down to just one trial, that if somebody requests a ride within 143 00:11:55,160 --> 00:12:02,030 our ridesharing app, that there's a 75% chance that a driver will arrive within 10 minutes for that 144 00:12:02,030 --> 00:12:03,500 particular ride. 145 00:12:03,500 --> 00:12:09,470 But of course, similar to some of the other distributions we've looked at, it's really nonsensical 146 00:12:09,470 --> 00:12:14,570 to think about one particular driver arriving 75% of the time. 147 00:12:14,570 --> 00:12:21,980 One particular driver is either going to arrive within 10 minutes, success or not failure. 148 00:12:21,980 --> 00:12:27,800 One particular driver will always take on one of those two values only and nothing in between. 149 00:12:27,800 --> 00:12:30,260 They will either succeed or fail. 150 00:12:30,260 --> 00:12:37,280 We can always code them as specifically a one or a zero, and every single individual driver we pick 151 00:12:37,280 --> 00:12:40,040 will always be a one or a zero. 152 00:12:40,190 --> 00:12:49,280 No one driver will ever fall in between zero or one as a 0.4 or a 0.637 or a 0.75. 153 00:12:49,280 --> 00:12:52,310 They will always be a zero or a one individually. 154 00:12:52,310 --> 00:12:58,520 And yet we're saying that the probability of success for every single driver is some value between zero 155 00:12:58,520 --> 00:12:59,090 and one. 156 00:12:59,090 --> 00:13:00,980 It is 0.75. 157 00:13:00,980 --> 00:13:07,760 So the whole goal here is just to understand that while every driver is only going to be success or 158 00:13:07,760 --> 00:13:12,590 failure, they're only going to be one or zero, they're only going to be on or off. 159 00:13:12,590 --> 00:13:19,190 They will only ever take on one of two values at the same time we associate with them this probability 160 00:13:19,190 --> 00:13:20,150 of success. 161 00:13:20,150 --> 00:13:25,700 That is some value in between those two exclusive discrete values. 162 00:13:25,700 --> 00:13:29,720 And every single ride has that same probability of success. 163 00:13:29,720 --> 00:13:33,530 P of 0.75. 164 00:13:33,530 --> 00:13:40,340 Even though each individual ride always gets coded as a failure of zero, the driver doesn't arrive 165 00:13:40,340 --> 00:13:43,640 within 10 minutes or a success of one. 166 00:13:43,640 --> 00:13:46,670 The driver does arrive within 10 minutes.