Remember that both Poisson and Binomial are discrete distributions.
Given:
Use:
or you can find the pmf of y and then the mean and variance.
....
- "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?