Question

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.

Answer #1

Calculate the quantity of interest please.
a) Let X,Y be jointly continuous random variables generated as
follows: Select X = x as a uniform random variable on [0,1]. Then,
select Y as a Gaussian random variable with mean x and variance 1.
Compute E[Y ].
b) Let X,Y be jointly Gaussian, with mean E[X] = E[Y ] = 0,
variances V ar[X] = 1,V ar[Y ] = 1 and covariance Cov[X,Y ] = 0.4.
Compute E[(X + 2Y )2].

Let X and Y denote be as follows: E(X) = 10, E(X2) =
125, E(Y) = 20, Var(Y) =100 , and Var(X+Y) = 155. Let W = 2X-Y and
let T = 4Y-3X. Find the covariance of W and T.

Let X and Y be independent and identically
distributed random variables with mean μ and variance
σ2. Find the following:
a) E[(X + 2)2]
b) Var(3X + 4)
c) E[(X - Y)2]
d) Cov{(X + Y), (X - Y)}

If X, Y are random variables with E(X) = 2, Var(X) = 3, E(Y) =
1, Var(Y) =2, ρX,Y = −0.5
(a) For Z = 3X − 1 find µZ, σZ.
(b) For T = 2X + Y find µT , σT
(c) U = X^3 find approximate values of µU , σU

Suppose that X and Y are two jointly continuous random variables
with joint PDF
??,(?, ?) =
??
??? 0 ≤ ? ≤ 1 ??? 0 ≤ ? ≤ √?
0
??ℎ??????
Compute and plot ??(?) and ??(?)
Are X and Y independent?
Compute and plot ??(?) and ???(?)
Compute E(X), Var(X), E(Y), Var(Y), Cov(X,Y), and
Cor.(X,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

Uncorrelated and Gaussian does not imply independent unless
jointly Gaussian. Let X ∼N(0,1) and Y = WX, where p(W = −1) = p(W =
1) = 0 .5. It is clear that X and Y are not independent, since Y is
a function of X. a. Show Y ∼N(0,1). b. Show cov[X,Y ]=0. Thus X and
Y are uncorrelated but dependent, even though they are Gaussian.
Hint: use the deﬁnition of covariance cov[X,Y]=E [XY] −E [X] E [Y ]
and...

8)
X1X1 and X2X2 are two random variables that are jointly
distributed such that E(X1)=8E(X1)=8, E(X2)=12E(X2)=12,
Var(X1)=2Var(X1)=2, Var(X2)=3Var(X2)=3 and
Cov(X1,X2)=−1Cov(X1,X2)=−1. Let Z=5X1−4X2−5Z=5X1−4X2−5.
Compute E(Z)E(Z).
Compute Var(Z)Var(Z).
Compute Corr(X1,X2)Corr(X1,X2).
Show all your working.

Let X and Y be jointly continuous random variables with joint
density function f(x, y) = c(y^2 − x^2 )e^(−2y) , −y ≤ x ≤ y, 0
< y < ∞.
(a) Find c so that f is a density function.
(b) Find the marginal densities of X and Y .
(c) Find the expected value of X

Consider two random variables X and Y such that E(X)=E(Y)=120,
Var(X)=14, Var(Y)=11, Cov(X,Y)=0.
Compute an upper bound to
P(|X−Y|>16)

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