How to find average of minimum value of two Poisson distributions?

Question

I have two Poisson distributions with parameters $\lambda 1$ and $\lambda 2$.
Now I want to find the average of minimum value of this two distributions.

I mean that if I derive two random numbers from these distributions and find the minimum of this two values, and repeat this experiment so many times, what would be the average of the minimum value.

Is this true to consider min ($\lambda 1$ , $\lambda2$ ) as the minimum value?

I appreciate your help. It is very important to me.

Thanks

Answer 1

For $i=1,2$ let $X_i$ be independent random variables Poisson-$\lambda_i$ distributed.

You seem to be asking:

"Is it true that in this case $\mathbb E\min(X_1,X_2)=\min(\mathbb EX_1,\mathbb EX_2)$?"

The answer is: "no".

It is evident that $\min(X_1,X_2)\leq X_1$ and $P(\min(X_1,X_2)<X_1)>0$.

These observations justify the conclusion that $\mathbb E\min(X_1,X_2)<\mathbb EX_1=\lambda_1$.

Likewise we find that $\mathbb E\min(X_1,X_2)<\mathbb EX_2=\lambda_2$.

This together implies that $$\mathbb E\min(X_1,X_2)<\min(\lambda_1,\lambda_2)$$