Bernoulli distributions and margin of error
Bernoulli Distribution Mean and Variance Formulas Bernoulli Distribution Mean and Variance Formulas
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- In the last video we figured out the mean, variance and
- standard deviation for our Bernoulli Distribution with
- specific numbers.
- What I want to do in this video is to generalize it.
- To figure out really the formulas for the mean and the
- variance of a Bernoulli Distribution if we don't have
- the actual numbers.
- If we just know that the probability of success is p
- and the probability a failure is 1 minus p.
- So let's look at this, let's look at a population where the
- probability of success-- we'll define success as 1-- as
- having a probability of p, and the probability of failure,
- the probability of failure is 1 minus p.
- Whatever this might be.
- And obviously, if you add these two up, if you view them
- as percentages, these are going to add up to 100%.
- Or if you add up these two values, they are
- going to add to 1.
- And that needs to be the case because these are the only two
- possibilities that can occur.
- If this is 60% chance of success there has to be a 40%
- chance of failure.
- 70% chance of success, 30% chance of failure.
- Now with this definition of this-- and this is the most
- general definition of a Bernoulli Distribution.
- It's really exactly what we did in the last video, I now
- want to calculate the expected value, which is the same thing
- as the mean of this distribution, and I also want
- to calculate the variance, which is the same thing as the
- expected squared distance of a value from the mean.
- So let's do that.
- So what is the mean over here?
- What is going to be the mean?
- Well that's just the probability weighted sum of
- the values that this could take on.
- So there is a 1 minus p probability that we get
- failure, that we get 0.
- So there's 1 minus p probability of
- getting 0, so times 0.
- And then there is a p probability of getting 1,
- plus p times 1.
- Well this is pretty easy to calculate.
- 0 times anything is 0.
- So that cancels out.
- And then p times 1 is just going to be p.
- So pretty straightforward.
- The mean, the expected value of this distribution, is p.
- And p might be here or something.
- So once again it's a value that you cannot actually take
- on in this distribution, which is interesting.
- But it is the expected value.
- Now what is going to be the variance?
- What is the variance of this distribution?
- Remember, that is the weighted sum of the squared distances
- from the mean.
- Now what's the probability that we get a 0?
- We already figured that out.
- There's a 1 minus p probability that we get a 0.
- So that is the probability part.
- And what is the squared distance from 0 to our mean?
- Well the squared distance from 0 to our mean-- let me write
- it over here-- it's going to be 0, that's the value we're
- taking on-- let me do that in blue since I already wrote the
- 0-- 0 minus our mean-- let me do this in a new
- color-- minus our mean.
- That's too similar to that orange.
- Let me do the mean in white.
- 0 minus our mean, which is p plus the probability that we
- get a 1, which is just p-- this is the squared distance,
- let me be very careful.
- It's the probability weighted sum of the squared distances
- from the mean.
- Now what's the distance-- now we've got a 1-- and what's the
- difference between 1 and the mean?
- It's 1 minus our mean, which is going to be p over here.
- And we're going to want to square this as well.
- This right here is going to be the variance.
- Now let's actually work this out.
- So this is going to be equal to 1 minus p.
- Now 0 minus p is going to be negative p.
- If you square it you're just going to get p squared.
- So it's going to be p squared.
- Then plus p times-- what's 1 minus p squared?
- 1 minus p squared is going to be 1 squared, which is just 1,
- minus 2 times the product of this.
- So this is going to be minus 2p right over here.
- And then plus negative p squared.
- So plus p squared just like that.
- And now let's multiply everything out.
- This is going to be, this term right over here is going to be
- p squared minus p to the third.
- And then this term over here, this whole thing over here, is
- going to be plus p times 1 is p.
- p times negative 2p is negative 2p squared.
- And then p times p squared is p to the third.
- Now we can simplify these.
- p to the third cancels out with p to the third.
- And then we have p squared minus 2p squared.
- So this right here becomes, you have this p right over
- here, so this is equal to p.
- And then when you add p squared to negative 2p squared
- you're left with negative p squared minus p squared.
- And if you want to factor a p out of this, this is going to
- be equal to p times, if you take p divided p you get a 1,
- p square divided by p is p.
- So p times 1 minus p, which is a pretty neat, clean formula.
- So our variance is p times 1 minus p.
- And if we want to take it to the next level and figure out
- the standard deviation, the standard deviation is just the
- square root of the variance, which is equal to the square
- root of p times 1 minus p.
- And we could even verify that this actually works for the
- example that we did up here.
- Our mean is p, the probability of success.
- We see that indeed it was, it was 0.6.
- And we know that our variance is essentially the probability
- of success times the probability of failure.
- That's our variance right over there.
- The probability of success in this example was 0.6,
- probability of failure was 0.4.
- You multiply the two, you get 0.24, which is exactly what we
- got in the last example.
- And if you take its square root for the standard
- deviation, which is what we do right here, it's 0.49.
- So hopefully you found that helpful, and we're going to
- build on this later on in some of our inferential statistics.
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