Discuss the below:
1. Let X and Y be independent normal rv's, each with mean mu and variance sigma^2. Use moment generating functions to show that X+Y and X-Y are independent normal rv's.
2. If X and Y are independent and M_X(t)=exp{2e^t-2} and M_Y(t)=(3/4 e^t + 1/4}^{10}. What is P(XY=0)?
3. Two dice are rolled and X is the sum. Compute M_X.
Problems on limit theorems
Let Phi(x)=P(Z
Response to the questions below may be expressed in terms of the function Phi(x).
4. Treating student test scores as i.i.d., in a test where the mean is 75 and variance is 25, what is the probability that a student will score between 65 and 85?
5. Fifty numbers are rounded off to the nearest integer and then summed. If the individual round-off errors are independent and uniformly distributed over (-0.5, 0.5), what is the probability that the resultant sum differs from the exact sum by more than 3?
6. A die is continually rolled until the total sum of all rolls exceeds 300. What is the prob that at least 80 rolls are necessary?
7. Compute P(X>120) for a Poisson rv with mean 100.