Let x and y be independent binomial random variables having parameters(n, p) and (m, p), respectively. let z = x+y; (a) argue that z has a binomial distribution with parameters(n+m, p) bywriting x and y as appropriate sums of bernoulli random variables.(b) validate the result in (a) by evaluating the necessary convolution.

See explanation below.

Step-by-step explanation:

Part a

For this case we can use the moment generating function for the bernoulli distribution for n trials, given by:

Wehre p is the probability of success and

Using this property we see that;

If we multiply the two generating functions we got:

And the mass function would be given by:

So we see that follows a binomial random variable with parameters (N+M, p)

Part b

For this case we are assuming that and for this case we can assume that for the proof.

We are interested on the random variable since the two random variables are independent we can write the probability mass function for Z like this:

And we can replace the mass function for X and Y

And we can rewrite this like that:

[We can take out the constant p:

And using properties of the binomial formula we can write this like that:

So then we see that