1. Assume that X and Y are two independent discrete random
variables and that X~N(0,1) and Y~N(µ,σ2).
a. Derive E(X3) and deduce that E[((Y-µ)/σ)3]
= 0
b. Derive P(X > 1.65). With µ = 0.5 and σ2 = 4.0,
find z such that P(((Y-µ)/σ) ≤ z) = 0.95. Does z depend on µ and/or
σ? Why
A: Suppose two random variables X and Y are independent and
identically distributed as standard normal. Specify the joint
probability density function f(x, y) of X and Y.
Next, suppose two random variables X and Y are independent and
identically distributed as Bernoulli with parameter 1 2 . Specify
the joint probability mass function f(x, y) of X and Y.
B: Consider a time series realization X = [10, 15, 23, 20, 19]
with a length of five-periods. Compute the...
Let X and Y be two independent random variables. X is a binomial
(25,0.4) and Y is a uniform (0,6). Let W=2X-Y and Z= 2X+Y.
a) Find the expected value of X, the expected value of Y, the
variance of X and the variance of Y.
b) Find the expected value of W.
c) Find the variance of W.
d) Find the covariance of Z and W.
d) Find the covariance of Z and W.
Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,
P(X = 1, Y = 0) = .3, P(X = 2, Y = 0) = .2 P(X = 0, Y = 1) = .2,
P(X = 1, Y = 1) = .2, P(X = 2, Y = 1) = 0.
a. Determine E(X) and E(Y ).
b. Find Cov(X, Y )
c. Find Cov(2X + 3Y, Y ).
9.8 Let X and Y be independent random variables with probability
distributions given by
P(X = 0) = P(X = 1) = 1/2 and P(Y = 0) = P(Y = 2) = 1/2 .
a. Compute the distribution of Z = X + Y .
b. Let Y˜ and Z˜ be independent random variables, where Y˜ has
the same distribution as Y , and Z˜ the same distribution as Z.
Compute the distribution of X˜ = Z˜ − Y
Let X and Y be two independent random variables, and g : R2
--> R an arbitrary bivariate function.
1) Suppose that X and Y are continuous with densities fX and fY
. Prove that for any y ? R withfY (y) > 0, the conditional
density of the random variable Z = g(X, Y ) given Y = y is the same
as the density of the random variable W = g(X, y).
2) Suppose that X and Y...