Probability mass function and joint density

1.A receiving depot receives a shipment of 100 generators, 5 of which are defective.

Four are selected at random without replacement for inspection. Let

*Y*

be the number of defectives.

(a) What is the probability mass function of

*Y *?

(b) Calculate *P*(1 *≤ Y ≤ *3).

2. Suppose the density of *X *is given by

$\displaystyle

f(x)=\frac{1}{4} xe^{-\frac{x}{2}} \quad x>0

$

0 otherwise

Use the moment generating function to calculate Var (*X*).

3.

Let

*X *and *Y *be independent normal random variables, each having parameters *μ *and *σ*2.

(a) Compute the joint density of *A *= *X *+ *Y *and *B *= *X − Y *.

(b) Are *A *and *B *independent?