let X1 and X2 be iid possion variable with mean 1. Then P[max( X1,X2)] is?

Question

let X1 and X2 be iid possion variable with mean 1. Then  P[max( X1,X2)] is?

anonymous · Answer

What is the question? Sorry, I'm having trouble reading it! :P

anonymous · Answer

Let $X=\max(X_1,X_2)$, then the cdf of $Y$ is given by
$$\begin{align*}F_X(x)&=P(X_1\le x,X_2\le x)\
&=P(X_1\le x)P(X_2\le x)&\text{since }X_1\text{ and }X_2\text{ are i.i.d.}\
&=F(x)F(x)&\text{by definition of c.d.f.}\
&=\left[F(x)\right]^2
\end{align*}$$
$P(X)=P(\max(X_1,X_2))$ is then given by the pdf of $X$:
$$f_X(x)=\frac{d}{dx}\left[F(x)\right]^2=2F(x)f(x)$$
So really, the question is a matter of multiplying the cdf and pdf of the Poisson distribution. I don't know them off the top of my head.