Mean and variance of Bernoulli distribution example Mean and Variance of Bernoulli Distribution Example
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
- 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
- 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
- It's the simplest case of the binomial distribution.
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