Hi, I'm revising for an upcoming test and have a past paper with answers but I can't understand where they are coming from. If anyone could try explaining this too me it would be greatly appreciated.

The number of eggs laid by an insect is a random variable

which has a poisson distribution with expectation

so

for

Independently for each egg laid there is probability

each that it develops into a male insect or a female insect. Let

denote the number of male offspring, and

the number of female offspring.

i) Find the joint probability mass function of

ii) Show that the marginal distribution of

, the number of male offspring, is Poisson with expectation

The answers I've been given are

i)

for non negative intergers

ii)For non-negative integers

,

as the sum is

If anyone can explain how you get to these answers it would be a massive help,

Cheers