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Practice Exam question. Not sure how to do this.

The number of offspring of an organism is a discrete random variable with mean $\mu$ and variance $\sigma^2$. Each of its offspring reproduce in the same manner. Find the expected number of offspring in the third generation and its variance.

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Let $O_i$ be the total number of offspring in generation $i$.

We know that $E[O_1]=\mu$, we also know that $E[O_2|O_1]=O_1\mu$, similarly, $E[O_3|O_2]=O_2\mu$

However, we don't want the conditional expected value, so we take expectations over the conditional expectations:

$E[E[O_2|O_1]]=E[O_2]=E[O_1\mu]=\mu E[O_1]=\mu^2$

Similarly, $E[E[O_3|O_2]]=E[O_3]=E[O_2]\mu=\mu^2 \mu = \mu^3$

We can do the same thing for the variance:

$Var[O_1]=\sigma^2$, $Var[O_2|O_1]=O_1\sigma^2$, similarly, $V[O_3|O_2]=O_2\sigma^2$

Now, we can take expected values over each conditional variance to get:

$E[Var[O_2|O_1]]=Var[O_2]=E[O_1\sigma^2]=\sigma^2 E[O_1]=\mu \sigma^2$

Similarly, $E[Var[O_3|O_2]]=Var[O_3]=E[O_2]\sigma^2=\mu^2\sigma^2$

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