Probability
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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.
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Solution Summary
This is a set of questions regarding probability, moment generating functions, and limit theorems.
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