# Solved – Probability mass function of product of two binomial variables

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
$$f(Z, n, p).$$

Contents

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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# Solved – Probability mass function of product of two binomial variables

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
$$f(Z, n, p).$$

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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