For each given p, let have a binomial distribution with parameters p and suppose that is itself binomially distributed with parameters and formulate as a random sum and show that has a binomial distribution with parameters and

See the proof below.

Step-by-step explanation:

Assuming this complete question: "For each given p, let Z have a binomial distribution with parameters p and N. Suppose that N is itself binomially distributed with parameters q and M. Formulate Z as a random sum and show that Z has a binomial distribution with parameters pq and M."

Solution to the problem

For this case we can assume that we have N independent variables with the following distribution:

bernoulli on this case with probability of success p, and all the N variables are independent distributed. We can define the random variable Z like this:

From the info given we know that

We need to proof that by the definition of binomial random variable then we need to show that:

The deduction is based on the definition of independent random variables, we can do this:

And for the variance of Z we can do this:

And if we take common factor we got:

And as we can see then we can conclude that