Consider a large population of families, and suppose that the number of children in the different families are independent Poisson random variables with mean $\lambda$. Show that the number of siblings of a randomly chosen child is also Poisson distributed with mean $\lambda$.
My approach:
For any random child, if it has $k$ siblings, it implies that its parent had $k+1$ children. Hence, if $S =$ no. of siblings and if $C =$ no. of children
I'm not sure how to proceed after this. I tried evaluating the mean of S, by computing
I'm not sure where I'm going wrong and how to proceed.
EDIT: Found an answer in one of the solution manuals. Basically,
The probability of choosing a child that has $j$ siblings is the fraction of total children that have $j$ siblings.
Now, if $Z$ is the total number of families, hence the total no. of children would be $\lambda Z$. Also, if $P(j+1)$ is the probability that a family has $j+1$ children, then the no. of families with $j+1$ children is $Z \cdot P(j+1)$. Also, each of this family has $(j+1)$ children, each of whom have $j$ siblings. Hence, there are in total
children each having $j$ siblings. This divided by total number of children gives the fraction of children with $j$ siblings, i.e.
Clearly my answer is wrong in the first step itself. However I'm not able to articulate why the initial step is incorrect.