00:01
Okay, so it says we've got yi for i equals one to n, and each yi follows a binomial distribution with parameters ni and pi.
00:21
And the question asks us for the distribution of the sum of all the y variables.
00:30
Now, the way that i'm gonna do this is i'm gonna write each yi as the sum over some bernoulli random variables.
00:42
And it will be the sum of j equals one to ni, which is the number of trials.
00:47
Remember, any binomial variable, if x is a binomial variable with parameters n and p, then x is just the sum of n bernoulli random variables, which take value one with probability p and value zero with probability one minus p.
01:04
And because these b, i, j are all independent because the yi are independent, then we can see that the sum of yi from i equals one to n is simply the sum of over i and j from one to, well, i should actually, so it's the sum from i equals one to n of the sum of j equals one to ni of b, i, j.
01:42
And so this is just the sum of all these bernoulli independent variables.
01:46
So this is just going to be binomial because it's the sum of independent bernoulli random variables...