I have two i.i.d. binomial variables $X$ and $Y$ with given $n$ and $p.$

What is probability mass function of $Z = X times Y$? I need pmf as function

$f(Z, n, p).$

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#### Best Answer

There are various ways you could write the mass function of this distribution. All of them will be messy, since they involve checking the possible products that give a stipulated value for the product variable. Here is the most obvious way to write the distribution.

Let $X, Y sim text{IID Bin}(n, p)$ and let $Z=XY$ be their product. For any integer $0 leqslant z leqslant n^2$ we define the set of pairs of values:

$$mathcal{S}(z) equiv { (x,y) in mathbb{N}_{0+}^2 mid max(x,y) leqslant n, xy=z }.$$

This is the set of all pairs of values within the support of the binomial that multiply to the value $z$. (Note that it will be an empty set for some values of $z$.) We then have:

$$begin{equation} begin{aligned} p_Z(z) equiv mathbb{P}(Z=z) &= mathbb{P}(XY=z) \[6pt] &= sum_{(x,y) in mathcal{S}(z)} text{Bin}(xmid n,p) cdot text{Bin}(ymid n, p) \[6pt] &= sum_{(x,y) in mathcal{S}(z)} {n choose x} {n choose y} cdot p^{x+y} (1-p)^{2n-x-y}. end{aligned} end{equation}$$

Computing this probability mass function requires you to find the set $mathcal{S}(z)$ for each $z$ in your support. The distribution has mean and variance:

$$mathbb{E}(Z) = (np)^2 quad quad quad quad quad mathbb{V}(Z) = (np)^2 [(1-p+np)^2 – (np)^2].$$

The distribution will be quite jagged, owing to the fact that it is the distribution of a product of discrete random variables. Notwithstanding its jagged distribution, as $n rightarrow infty$ you will have convergence in probability to $Z/n^2 rightarrow p^2$.

**Implementation in R:** The easiest way to code this mass function is to first create a matrix of joint probabilities for the underlying random variables $X$ and $Y$, and then allocate each of these probabilities to the appropriate resulting product value. In the code below I will create a function

`dprodbinom`

which is a vectorised function for the probability mass function of this "product-binomial" distribution.`#Create function for PMF of the product-binomial distribution dprodbinom <- function(Z, size, prob, log = FALSE) { #Check input vector is numeric if (!is.numeric(Z)) { stop('Error: Input values are not numeric'); } #Set parameters n <- size; p <- prob; #Generate matrix of joint probabilities SS <- matrix(-Inf, nrow = n+1, ncol = n+1); XX <- dbinom(0:n, size = n, prob = p, log = TRUE); for (x in 0:n) { for (y in 0:n) { SS[x+1, y+1] <- XX[x+1] + XX[y+1]; } } #Compute the log-mass function of the product random variable LOGPMF <- rep(-Inf, n^2+1); for (x in 0:n) { for (y in 0:n) { LOGPMF[x*y+1] <- matrixStats::logSumExp(c(LOGPMF[x*y+1], SS[x+1, y+1])); } } #Generate the output vector OUT <- rep(-Inf, length(Z)); for (i in 1:length(Z)) { if (Z[i] %in% 0:(n^2)) { OUT[i] <- LOGPMF[Z[i]+1]; } } #Give the output of the function if (log) { OUT } else { exp(OUT) } } `

We can now easily generate and plot the probability mass function of this distribution. For example, with $n=10$ and $p = 0.6$ we obtain the following probability mass function. As you can see, it is quite jagged, owing to the fact that the product values are distributed in a lagged pattern over the joint values of the underlying random variables.

`#Load required libraries library(matrixStats); library(ggplot2); #Generate the mass function n <- 10; p <- 0.6; PMF <- dprodbinom(0:100, size = n, prob = p, log = FALSE); #Plot the mass function THEME <- theme(plot.title = element_text(hjust = 0.5, size = 14, face = 'bold'), plot.subtitle = element_text(hjust = 0.5, face = 'bold')); DATA <- data.frame(Value = 0:100, Probability = PMF); FIGURE <- ggplot(aes(x = Value, y = Probability), data = DATA) + geom_bar(stat = 'identity', colour = 'blue') + THEME + ggtitle('Product-binomial probability mass function') + labs(subtitle = paste0('(n = ', n, ', p = ', p, ')')); FIGURE; `

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