# expected value and variance

• Mar 4th 2010, 01:06 PM
brogers
expected value and variance
A certain organism produces an average of $\mu$ offsprings with variance $\sigma ^2$. Each offspring produces likewise. What would be the expected number of offspring and its variance in the third generation?
If $X_{i}$ = # of offspring in ith generation, I figured $E(X_{3})= \mu ^2$ and $Var(X_{3}) = \mu ^2 \sigma ^2$
I'm pretty sure of expected, but not that sure of variance. I used the fact that if $Y= a+bX$ then $Var(Y)=b^{2}Var(X)$
Can somebody confirm this? Thanks.
• Mar 4th 2010, 01:29 PM
Moo
Hello,

This is similar to a Galton-Watson process (if it is assumed that the production of offsprings are independent).
And assuming that the "parents" die once they give their offspring, we have $X_{n+1}=\sum_{k=1}^{X_n} \xi_{k,n+1}$, where $(\xi_{k,n})_n$ is an iid sequence with mean $\mu$ and variance $\sigma^2$

And you're looking for $E[X_3]$ and $Var[X_3]$ (where $X_0$ - or $X_1$ ? - is 1)

It's easy to compute $E[X_{n+1}|X_n]$ (using basic properties of conditional expectations), and it gives the result you got by taking the expectation again.

And for the variance, I suggest you use the law of total variance : Law of total variance - Wikipedia, the free encyclopedia

If you have trouble, let us know (Nerd)