Question

Suppose that X ∼ Geom(p) and Y ∼ Geom(r) are independent. Find the probability P (X < Y ).

Answer 1

$X,Y$ have joint density

$f_{X,Y}(x,y)=f_X(x)f_Y(y)=\begin{cases}pr(1-p)^x(1-r)^y&\text{for }x\ge0,y\ge0\\0&\text{otherwise}\end{cases}$

We can find $P(X$ by integrating the joint density over the region in $\Bbb R^2$ where $x$ in the first quadrant, i.e. above the line $y=x$:$P(X$

using the fact that

$a^n=e^{n\ln a}\implies\displaystyle\int a^n\,\mathrm dn=\frac1{\ln a}e^{n\ln a}+C=\frac1{\ln a}a^n+C$

Then using the same property,

$P(X$