Question

Let *X* and *Y* be independent and identically
distributed random variables with mean μ and variance
σ^{2}. Find the following:

a) *E*[(*X* + 2)^{2}]

b) Var(3*X* + 4)

c) *E*[(*X* - *Y*)^{2}]

d) Cov{(*X* + *Y*), (*X* - *Y*)}

Answer #1

Suppose that X1, X2, . . . , Xn are independent identically
distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]

Let X and Y be jointly distributed random variables with means,
E(X) = 1, E(Y) = 0, variances, Var(X) = 4, Var(Y ) = 5, and
covariance, Cov(X, Y ) = 2.
Let U = 3X-Y +2 and W = 2X + Y . Obtain the following
expectations:
A.) Var(U):
B.) Var(W):
C. Cov(U,W):
ans should be 29, 29, 21 but I need help showing how to
solve.

Suppose that X1,X2 and X3 are independent random variables with
common mean E(Xi) = μ and variance Var(Xi) = σ2. Let V= X2−X3 and W
= X1− 2X2 + X3.
(a) Find E(V) and E(W).
(b) Find Var(V) and Var(W).
(c) Find Cov(V,W).
(d) Find the correlation coefficient ρ(V,W). Are V and W
independent?

Let
x1, x2 x3 ....be a sequence of independent and identically
distributed random variables, each having finite mean E[xi] and
variance Var（xi）.
a）calculate the var （x1+x2）
b）calculate the var（E[xi]）
c） if n-> infinite, what is Var（E[xi]）？

7.
Let X and Y be two independent and identically distributed
random variables with expected value 1 and variance 2.56.
(i) Find a non-trivial upper bound for
P(| X + Y -2 | >= 1)
(ii) Now suppose that X and Y are independent and identically
distributed N(1;2.56) random variables. What is P(|X+Y=2| >= 1)
exactly? Briefly, state your reasoning.
(iii) Why is the upper bound you obtained in Part (i) so
different from the exact probability you obtained in...

Suppose X1, X2, X3, and
X4 are independent and identically distributed random
variables with mean 10 and variance 16. in addition, Suppose that
Y1, Y2, Y3, Y4, and
Y5are independent and identically distributed random
variables with mean 15 and variance 25. Suppose further that
X1, X2, X3, and X4 and
Y1, Y2, Y3, Y4, and
Y5are independent. Find Cov[bar{X} + bar{Y} + 10,
2bar{X} - bar{Y}], where bar{X} is the sample mean of
X1, X2, X3, and X4 and
bar{Y}...

Let X and Y be independent and normally distributed random
variables with waiting values E (X) = 3, E (Y) = 4 and variances V
(X) = 2 and V (Y) = 3.
a) Determine the expected value and variance for 2X-Y
Waiting value µ = Variance σ2 = σ 2 =
b) Determine the expected value and variance for ln (1 + X
2)
c) Determine the expected value and variance for X / Y

Let X, Y, and Z be independent and identically distributed
discrete random variables, with each having a probability
distribution that puts a mass of 1/4 on the number 0, a mass of 1/4
at 1, and a mass of 1/2 at 2.
a. Compute the moment generating function for S= X+Y+Z
b. Use the MGF from part a to compute the second moment of S,
E(S^2)
c. Compute the second moment of S in a completely different way,
by expanding...

Let X and Y be independent, identically distributed standard
uniform random variables. Compute the probability density function
of XY .

Problem 3. Let Y1, Y2, and Y3 be independent, identically
distributed random variables from a population with mean µ = 12 and
variance σ 2 = 192. Let Y¯ = 1/3 (Y1 + Y2 +
Y3) denote the average of these three random
variables.
A. What is the expected value of Y¯, i.e., E(Y¯ ) =? Is Y¯ an
unbiased estimator of µ?
B. What is the variance of Y¯, i.e, V ar(Y¯ ) =?
C. Consider a different estimator...

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