Finding P{X = Y} where X~Geo(P), Y| X = i ~Binomial(i,p)

Question

I need assistance figuring out how to address a specific question. I don't need the complete answer, just some guidance would be appreciated.

Let $X$ be a geometric random variable with $P$ as the parameter. let $Y$, given $X = i$, a binomial random variable with parameters (i, p).

find $P\{X = Y\}$.

I literally have no idea how to attack this.

Answer 1

Note that $$ P(X = Y) = \sum_{i = 0}^{\infty} P(X =i, Y= i) = \sum_{i = 0}^{\infty} P(Y = i \mid X = i) P(X = i), $$ but $P(Y = i \mid X = i) = \binom{i}{i}p^i(1-p)^{i-i}= p^i$, since $Y \mid X = i$ has binomial distribution. $P(X = i) = (1-p)^{i} p$ since $X$ has geometric distribution. Replacing this results you will get a convergent series.

Answer 2

Use the law of total probability: $$P\{X=Y\} = \sum_i P\{X=Y \mid X=i\} P\{X=i\}$$