let $\mathit X\sim Geo(p) $ and $\mathit Y\sim Geo(3p)$ independent random variables and $\mathit M=Max\{X,Y \} $.
Find $\Bbb E[ M ]$?
We know that $\Bbb E[ Y ]=\frac{1}{3p} $ and $\Bbb E[ X ]=\frac{1}{p}$ , so we know for sure that $\Bbb E[ M ]\gt\frac{1}{p} $ since we are taking the maximum.
But how do we calculate the exact value?