Prob and Stat Homework 9
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Prob and Stat
Homework 9
Problem 1
Suppose (X, Y) has a joint pdf f (x, y) = e-x I(0<y<x<o} . Find
(a) fX (x) and fY (y)
(b) fX|Y (x|y) and fY |X (y|x)
(c) E(X|Y) and E(Y |X)
(d) Var(X|Y) and Var(Y |X)
Problem 2
Suppose we have created a portfolio, where we invest a unit in bond A and 1 [ a unit in bond B. Let X and Y be the return of one unit bond A and one unit bond B respectively. Denote the mean and variance as µX , µY , and σX(2) , σY(2) . Assume ρ is the correlation coefficient between X and Y . What is the average return and risk of the portfolio? Find a* that minimize the investment risk. (Hint: Risk can be considered as variance of return from the whole portfolio.)
Problem 3
Suppose (X, Y) has density af1 (x)g1 (y)+(1 [a)f2 (x)g2 (y), where a is a constant between 0 and 1, f1 , f2 , g1 , g2 are univariate densities with means µ 1 , µ2 , ν1 , ν2 .
(a) Show that the marginal density of X is fX (x) = af1 (x) + (1 [ a)f2 (x). (b) Show that X and Y are independent if and only if
[f1 (x) [ f2 (x)][g1 (y) [ g2 (y)] = 0.
(c) Show that cov(X, Y) = a(1 [ a)(µ1 [ µ2 )(ν1 [ ν2 ).
Problem 4
For any two random variables X and Y with finite variances, prove that
(a) X and Y [ E (Y | X) are uncorrelated.
(b) var[Y [ E (Y | X)] = E[var(Y | X)].
Problem 5
Write a summary of Chapter 5. You can design your own summary style.
2021-12-25