# Thread: Trouble understanding conditional expectation.

1. ## Trouble understanding conditional expectation.

• "A Poisson number of Bernoulli trials." Fix λ > 0 and 0 < p < 1, and let N be Poisson(λ). Assume that the conditional distribution of Y, given that N = n, is binomial(n, p) for n = 0, 1, ..., Find the mean and variance of Y by conditioning on N.

• "Conditional expectation given a sum." Fix n ≥ 2, let X1, . . . , Xn be n i.i.d. discrete random variables with finite expectation, and put Sn := X1 + X2· · · + Xn. Show that E[X1 | Sn] = Sn/n.

I am having trouble understanding conditional expectation; the textbook I have doesn't do a very good job explaining it. Does anyone know of any resources online that can help me get a better grasp and understanding this?

2. Remember that both Poisson and Binomial are discrete distributions.

Given: $Y|N \sim \mbox{Binomial}(n,p)$

$N \sim \mbox{Poisson}(\lambda)$

Use: $E[E[Y|N]] = E[Y]$

or you can find the pmf of y and then the mean and variance.

$\displaystyle P(Y=y)=\sum_{n=0}^{\infty} f(y,n)$

$\displaystyle P(Y=y)=\sum_{n=0}^{\infty} f(Y|N=n) \cdot f(n)$

$\displaystyle P(Y=y)= \sum_{n=0}^{\infty} \dbinom{n}{y} p^y(1-p)^{n-y} \cdot \dfrac{e^{-\lambda} {\lambda}^n}{n!}$

....