Question

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].

Answer #1

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.

Let X and Y be continuous random variables with joint
distribution function F(x, y), and let g(X, Y ) and h(X, Y ) be
functions of X and Y . Prove the following:
(a) E[cg(X, Y )] = cE[g(X, Y )].
(b) E[g(X, Y ) + h(X, Y )] = E[g(X, Y )] + E[h(X, Y )].
(c) V ar(a + X) = V ar(X).
(d) V ar(aX) = a 2V ar(X).
(e) V ar(aX + bY ) = a...

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)

True or false? and explain please.
(i) Let Z = 2X + 3. Then, Cov(Z,Y ) = 2Cov(X,Y ).
(j) If X,Y are uncorrelated, and Z = −5−Y + X , then V ar(Z) = V
ar(X) + V ar(Y ).
(k) Let X,Y be uncorrelated random variables (Cov(X,Y ) = 0) with
variance 1 each. Let A = 3X + Y . Then, V ar(A) = 4.
(l) If X and Y are uncorrelated (meaning that Cov[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...

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

Let continuous random variables X, Y be jointly continuous, with
the following joint distribution fXY(x,y) =
e-x-y for x≥0, y≥0 and fXY(x,y) = 0
otherwise.
1) Sketch the area where fXY(x,y) is non-zero on
x-y plane.
2) Compute the conditional PDF of Y given X=x for each
nonnegative x.
3) Use the results above to compute E(Y∣X=x) for each
nonnegative x.
4) Use total expectation formula E(E(Y∣X))=E(Y) to find
expected value of Y.

Let random variables X and Y follow a bivariate Gaussian
distribution, where X and Y are independent and Cov(X,Y) = 0.
Show that Y|X ~ Normal(E[Y|X], V[Y|X]). What are E[Y|X] and
V[Y|X]?

Suppose X and Y are continuous random variables with joint
density function fX;Y (x; y) = x + y on the square [0; 3] x [0; 3].
Compute E[X], E[Y], E[X2 + Y2], and Cov(3X -
4; 2Y +3).

Suppose X & Y are jointly continuous-type random variables
with the following joint CDF:
If u>=0 and v>=0:
F_X,Y(u,v) = {
min(1-e^-u,1-e^-v); if 0<=u<1, OR 0<=v<1
1; else
}
If u<0 or v<0: F_X,Y(u,v) = 0.
What is E[X]?

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