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# Probability

Problems on moment generating functions

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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

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Let Phi(x)=P(Z<x) where Z is the standard normal rv.

Answers 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.

Hint: think of X as the sum of 100 independent Poisson

rv's each with mean 1.