# Analytical expression for variance of nested binomials?

Hi all,

I want to compute the variance of a variable that is defined at each step as a recursion of binomials in the following way:

A=1

B=Bin(1,A)*Bin(1,p)

C=Bin(1,B)*Bin(1,p)

D=Bin(1,C)*Bin(1,p), etc...

Generalized form:

X(T+1)=Bin(1,X(T))*Bin(1,p);

Question: variance(X(T))=?

maybe it is too easy, but I would really appreciate some help...

It seems each $X(T)$ is Bernoulli, that is $X(T)=0$ or $1$ almost surely. As such, the variance of $X(T)$ is $a(T)(1-a(T))$ where $a(T)=E(X(T))$. But $a(0)=1$ and $a(T+1)=a(T)p$ hence $a(T)=p^T$ and the variance of $X(T)$ is $p^T(1-p^T)$.