Help! I'll buy you Yogurtland ;___;

Apr 03, 2010 22:30

Or Starbucks...! D: Italics represents what I'm having trouble on. I seriously will make this up to you if you help me. And... if you are good at this stuff, might you be interested in a part time tutoring job? :3;;;;

1. X is a Bernoulli random variable with Pr(X=1) = 0.99. Y is distributed N(0,1), and W is distributed N(0, 100). Let S = XY + (1-X)W. (That is, S = Y when X = 1, and S = W when X = 0)

a. Show that E(Y^2) = 1 and E(W^2) = 100.
b. Show that E(Y^3) = 0 and E(W^3) = 0. (Hint: What is the skewness for a symmetric distribution?)
c. Show that E(Y^4) = 3 and E(W^4) = 3 x 100^2. (Hint: Use the fact that the kurtosis is 3 for a normal distribution.)
d. Show that E(S), E(S^2), E(S^3) and E(S^4). (Hint: Use the law of iterated expectations conditioning on X=0 and X=1.)
e. Derive the skewness and kurtosis for X.

2. In any year, the weather can inflict storm damage to a home. From year to year, the damage is random. Let Y denote the dollar value of damage in any given year. Suppose that in 95% of the years Y = $0 but in 5% of the years Y = $20,000.

a. What is the mean and standard deviation of the damage in any year?
b. Consider an "insurance pool" of 100 people whose homes are sufficiently dispersed so that, in any year, the damage to different homes can be viewed as independently distributed random variables. Let Y-bar denote the average damage to these 100 homes in a year. (i) What is the expected value of the average damage Y-bar? (ii) What is the probability that Y-bar exceeds $2000? * I have a tentative answer but I have a feeling it's wrong.

3. This exercise provides an example of a pair of random variables X and Y for which the conditional mean of Y given X depends on X but corr(X,Y) = 0. Let X and Z be two independently distributed standard normal random variables, and let Y = X^2 + Z.

a. Show that E(Y|X) = X^2
b. Show that mu(y) = 1.
c. Show that E(XY) = 0. (Hint: Use the fact that the odd moments of a standard normal random variable are all zero)
d. Show that cov(X,Y) = 0 and thus corr(X,Y) = 0.

help, school

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