Conditional Expectation of Offspring in simplified Branching process

Question

Suppose $X_{0}, X_{1} ,....$ is a branching process who has an offspring distribution mean of $\mu$

Let

$$Y_{n}=\frac{X_{n}}{\mu^{n}}$$

I want to show that

$$E[Y_{n+1}|Y_{n}]=Y_{n}$$

Well,

I know that $E[X_{0}]=\mu$

and that $E[X_{n}]=\mu^{n}$

hence $E[Y_{n}]=E[Y_{n+1}]=1$

Now should I simply apply law of total expectation or some other basic? Or is there some key ideas I am missing. It is also possible I made mistakes in my reasoning above.

But I am really not sure, it seems that Y_{n} represents the actual size of the nth generation divided by the expected population of the nth generation

Answer 1

You can just plug the definition of $Y$ into $E[Y_{n+1} | Y_n = y]$, so you get $$E\left[\frac{X_{n+1}}{\mu^{n+1}} \left\vert \frac{X_n}{\mu^n}=y\right.\right]=\frac{1}{\mu^{n+1}}E\left[ X_{n+1}| X_n=\mu^n y\right].$$

Can you finish the rest?