# Thread: expected value and variance

1. ## expected value and variance

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

2. 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 $\displaystyle X_{n+1}=\sum_{k=1}^{X_n} \xi_{k,n+1}$, where $\displaystyle (\xi_{k,n})_n$ is an iid sequence with mean $\displaystyle \mu$ and variance $\displaystyle \sigma^2$

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

It's easy to compute $\displaystyle E[X_{n+1}|X_n]$ (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