Binomial mean and standard deviation formulas
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Mean and variance of Bernoulli distribution example
Let's say that I'm able to go out and survey every single member of a population, which we know is not normally practical, but I'm able to do it. And I ask each of them, what do you think of the president? And I ask them, and there's only two options, they can either have an unfavorable rating or they could have a favorable rating. And let's say after I survey every single member of this population, 40% have an unfavorable rating and 60% have a favorable rating. So if I were to draw the probability distribution, and it's going to be a discrete one because there's only two values that any person can take on. They could either have an unfavorable view or they could have a favorable view. And 40% have an unfavorable view, and let me color code this a little bit. So this is the 40% right over here, so 0.4 or maybe I'll just write 40% right over there. And then 60% have a favorable view. Let me color code this. 60% have a favorable view. And notice these two numbers add up to 100% because everyone had to pick between these two options. Now if I were to go and ask you to pick a random member of that population and say what is the expected favorability rating of that member, what would it be? Or another way to think about it is what is the mean of this distribution? And for a discrete distribution like this, your mean or you're expected value is just going to be the probability weighted sum of the different values that your distribution can take on. Now the way I've written it right here, you can't take a probability weighted sum of u and f-- you can't say 40% times u plus 60% times f, you won't get any type of a number. So what we're going to do is define u and f to be some type of value. So let's say that u is 0 and f is 1. And now the notion of taking a probability weighted sum makes some sense. So that mean, or you could say the mean, I'll say the mean of this distribution it's going to be 0.4-- that's this probability right here times 0 plus 0.6 times 1, which is going to be equal to-- this is just going to be 0.6 times 1 is 0.6. So clearly, no individual can take on the value of 0.6. No one can tell you I 60% am favorable and 40% am unfavorable. Everyone has to pick either favorable or unfavorable. So you will never actually find someone who has a 0.6 favorability value. It'll either be a 1 or a 0. So this is an interesting case where the mean or the expected value is not a value that the distribution can actually take on. It's a value some place over here that obviously cannot happen. But this is the mean, this is the expected value. And the reason why that makes sense is if you surveyed 100 people, you'd multiply 100 times this number, you would expect 60 people to say yes, or if you'd summed them all up, 60 would say yes, and then 40 would say 0. You sum them all up, you would get 60% saying yes, and that's exactly what our population distribution told us. Now what is the variance? What is the variance of this population right over here? So the variance-- let me write it over here, let me pick a new color-- the variance is just-- you could view it as the probability weighted sum of the squared distances from the mean, or the expected value of the squared distances from the mean. So what's that going to be? Well there's two different values that anything can take on. You can either have a 0 or you could either have a 1. The probability that you get a 0 is 0.4-- so there's a 0.4 probability that you get a 0. And if you get a 0 what's the distance from 0 to the mean? The distance from 0 to the mean is 0 minus 0.6, or I can even say 0.6 minus 0-- same thing because we're going to square it-- 0 minus 0.6 squared-- remember, the variance is the weighted sum of the squared distances. So this is the difference between 0 and the mean. And then plus, there's a 0.6 chance that you get a 1. And the difference between 1 and 0.6, 1 and our mean, 0.6, is that. And then we are also going to square this over here. Now what is this value going to be? This is going to be 0.4 times 0.6 squared-- this is 0.4 times point-- because 0 minus 0.6 is negative 0.6. If you square it you get positive 0.36. So this value right here-- I'm going to color code it. This value right here is times 0.36. And then this value right here-- let me do this in another-- so then we're going to have plus 0.6 times 1 minus 0.6 squared. Now 1 minus 0.6 is 0.4. 0.4 squared is 0.16. So let me do this. So this value right here is going to be 0.16. So let me get my calculator out to actually calculate these values. So this is going to be 0.4 times 0.36, plus 0.6 times 0.16, which is equal to 0.24. So our standard deviation of this distribution is 0.24. Or if you want to think about the variance of this distribution is 0.24 and the standard deviation of this distribution, which is just the square root of this, the standard deviation of this distribution is going to be the square root of 0.24, and let's calculate what that is. That is going to be-- let's take the square root of 0.24, which is equal to 0.48-- well I'll just round it up-- 0.49. So this is equal to 0.49. So if you were look at this distribution, the mean of this distribution is 0.6. So 0.6 is the mean. And the standard deviation is 0.5. So the standard deviation is-- so it's actually out here-- because if you go add one standard deviation you're almost getting to 1.1, so this is one standard deviation above, and then one standard deviation below gets you right about here. And that kind of makes sense. It's hard to kind of have a good intuition for a discrete distribution because you really can't take on those values, but it makes sense that the distribution is skewed to the right over here. Anyway, I did this example with particular numbers because I wanted to show you why this distribution is useful. In the next video I'll do these with just general numbers where this is going to be p, where this is the probability of success and this is 1 minus p, which is the probability of failure. And then we'll come up with general formulas for the mean and variance and standard deviation of this distribution, which is actually called the Bernoulli Distribution. It's the simplest case of the binomial distribution.