Question

Helmut goes fishing every Sunday. The number of fish he catches follows a Poisson distribution.
On a proportion π of the days he goes fishing, he does not catch anything. He makes it a rule
to take home the first and then every other fish that he catches (i.e. the first, third, fifth, and
so on).
Using a Poisson distribution, find the mean number of fish he catches.

Answer 1

Let

X be the total number of fishes he catches

Y is when he goes fishing

now ,we have

X|Y follows the poisson distribution

hence,mean number of fishes he catches is given by

E(X)= E(E(X|Y)) =   π *     {Assuming is mean for poisson distribution}