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Related to Probability mass function of product of two binomial variables

I am doing a Power Analysis for Sample Size determination. I have two binomial distributions: X ~ Bin(n,p) and Y ~ Bin(m,q). Typically, n is the number of previous tests n > 100, and p is the "Success Rate/Reliability", and m is the proposed number of new tests m < 15, with q being the "Reduced Success Rate/Degraded Reliability" such that q < p.

If I was to approach the problem, it should be as easy as obtaining a critical value from X and finding Beta/Power with that critical value in Y.

 crit = qbinom( alpha, n, p)
 beta = pbinom( crit, m, q)
 power = 1 - beta

But the problem that arises is the huge difference between the distributions. Any critical value from X will essentially always yield a Power = 1 in Y.

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That's where I've seen a product of two Binomials occur. In the following:

 space = expand.grid(0:n, 0:m)
 i = space[,1] 
 j = space[,2]
 null.dist = dbinom(i, n, p) * dbinom(j, m, p)
 alt.dist = dbinom(i, n, p) * dbinom(j, m, q)

And further calculations are made to obtain power from these distributions. My concern and questions is that the product of the two binomially distributed variables forms a new distribution altogether. All I have been able to discover is that E(XY) = E(X)E(Y) but I've encountered a hurdle here that I can't jump.

Is the product of two Binomial Random Variables also a Binomial Random Variable? Or something new?

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    $\begingroup$ You can see that it's not Binomial by considering the support. For instance, the product of two integers in the set $\{0,1,2\}$ can have the values 0, 1, 2, or 4, but never 3, whereas all (nondegenerate) Binomial$(n)$ distributions have positive chances for all the integers from 0 through $n.$ You therefore might find it more helpful to articulate your actual statistical problem rather than pursuing this question of what the product of two Binomial variables is. $\endgroup$
    – whuber ♦
    Commented Jan 6, 2021 at 21:32
  • $\begingroup$ The dbinom function doesn't return an integer, but the probability mass of that integer occurring within the distribution. i.e. dbinom(180,200,.9) = .094 I suppose I have poor wording here though. I'm essentially multiplying the pmf of X and Y together. $\endgroup$
    – Sumner18
    Commented Jan 6, 2021 at 21:40
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    $\begingroup$ @whuber is not confused about your question, he is simply telling you that the product cannot possibly be binomially distributed (by giving a counterexample in which the support does not reflect a binomial). The distribution will not have a closed form solution. Even in the special case where X and Y are independent (see en.wikipedia.org/wiki/Product_distribution) the distribution will be difficult to compute. However, from your description it does not seem like X and Y are independent. You can easily get an empirical estimate of the pmf through simulation though. $\endgroup$
    – bdeonovic
    Commented Jan 6, 2021 at 21:42

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