1 if a random variable x is distributed chi

Question

1. If a random variable X is distributed chi square with n degrees of freedom then the expected value of X is n. Show that this is true.

2. Suppose we have a random sample X1, X2, ...Xn from a distribution with mean µ and variance σ². Let's compare two estimators in terms of their relative efficiency. The first estimator is the sample mean Xbar. The second estimator is the second observation in the sample, X2. Which one is more efficient? (hint: calculate the expected value and variance of both)

3. You are given the following information about the joint probability distribution of X and Y:

	Y=1	Y=2	Y=3
X=0	0.25	0.1	0.05
X=1	0.3	0.2	0.1

Determine the following:

1. P(X=0|Y=2)
2. P(Y=3)
3. E(X|Y=1)
4. E(X)
5. Compare the answers in part c) and d). Do you get the same answer? Why or why not?