What is the expectation of the number of great-grandsons a cell have?

[maria clara]

Homework Statement

A cell diverges into X new cells. Each of them reproduces in the same manner. X is a geometric random variable with success parameter of 0.25.
What is the expectation of the number of great-grandsons a cell have?

2. The attempt at a solution
I thought about using the formula EX=EEX|Y somehow, but it wasn't very useful:
If the number of children is C
and the number of grandsons is G
and the number of great-grandsons is N
then
N|G=GX
G|C=CX
C=X

here I got stuck, is it the right direction at all?
thanks..:)

 

[cellotim]

The answer is simpler than that. This is an example of a Galton-Watson type of branching process.

 

[maria clara]

thanks

but I've never heard of this process, not in our lectures, at least...
is there any other way?

 

[cellotim]

Let $$Z_n$$ be the number of offspring for generation n. Prove the recursion relation $$E[Z_n] = E[Z_1]E[Z_{n-1}]$$, where $$Z_0=1$$ is the first generation. (Great-grandsons occurs at generation n=3.) You know that $$Z_1$$ is a geometrically distributed random variable as given and can find $$E[Z_1]$$. Now, note that the expected value of the number of offspring at generation n-1 will be the same as the expected value of the number of offspring that each "son" has at generation n. (It helps to draw a family tree.) Therefore, the number of offspring $$Z_n$$ is a sum of $$Z_1$$ random variables each with probability distribution of $$Z_{n-1}$$. You should now be able to prove the recursion.