Consider a sequence of independent Bernoulli trials, each of which is a success with probability p. Let X1 be the number of failures preceding the first success, and let X2 be the number of failures between the first two successes. Find the joint mass function of X1 and X2.

The joint probability mass function is a function that completely characterizes the distribution of a discrete random vector. It expresses the likelihood that the random vector's realization will be the same as that point when assessed at a specific place.

The number of failures preceding the first success is X₁

The number of failures between the first two successes is X₂

Let p be the probability of success.

Let i be the number of failures that occurs before the first success.

Let j be the number of failures that occurs between the first two successes.

The joint probability mass function of X₁ and X₂is$P(X_1=k,X_2=l)=P(X_1=k)P(X_2=l)={(1-p)}^k.p.{(1-p)}^l.p$

Therefore, the joint probability mass function of X₁and X₂is$P(X_1=k,X_2=l)={(1-p)}^k{(1-p)}^l{p}^2$