Product distribution

A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product is a product distribution. The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios. Many of these distributions are described in Melvin D. Springer's book from 1979 The Algebra of Random Variables. If X {displaystyle X} and Y {displaystyle Y} are two independent, continuous random variables, described by probability density functions f X {displaystyle f_{X}} and f Y {displaystyle f_{Y}} then the probability density function of Z = X Y {displaystyle Z=XY} is We first write the cumulative distribution function of Z {displaystyle Z} starting with its definition We find the desired probability density function by taking the derivative of both sides with respect to z {displaystyle z} . Since on the right hand side, z {displaystyle z} appears only in the integration limits, the derivative is easily performed using the fundamental theorem of calculus and the chain rule. (Note the negative sign that is needed when the variable occurs in the lower limit of the integration.)