Perform independent bernoulli trials, each of which is a success with probability $p$. Let $X_1$ be the number of failures preceding the first success and let $X_2$ be the number of failures between the first two successes. Find the joint mass function of $X_1$ and $X_2$.
Attempt
We have that
$$ P(X_1 = n) = (1-p)^n p $$
and
$$ P(X_2 = m ) = (1-p)^m p^2 $$
Since the trials are independent, we have
$$ P(X_1=n, X_2=m) = (1-p)^n (1-p)^m p^3 $$
Is this correct?
Best Answer
That would be the probability for $m$ consecutive failures and then two successes.
You don't want that.
You want the probability that, after the first success does happen (whenever it does), there are $m$ consecutive failures and then one success (the second).
$$ P(X_2 = m ) = (1-p)^m p^1 \qquad\color{green}\checkmark$$
All else is okay.