Question

Let
X & Y be two continuous random variables with joint pdf:

fXY(X,Y) = { 2 x+y =< 1, x >0, y>0

{ 0 otherwise

find Cov(X,Y) and ρX,Y

Answer #1

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.

Consider two random variables, X and Y, with joint PDF
fxy(x,y)=e-2|y-x^2|-x x>=0
, y can be any value
fxy(x,y)=0 otherwise
(1) Determine fY|X(y|x)
(2)Determine E[Y|X=x]

Let X and Y be continuous random variable with joint pdf
f(x,y) = y/144 if 0 < 4x < y < 12 and
0 otherwise
Find Cov (X,Y).

Suppose X and Y are continuous random variables with joint pdf
f(x,y) = 2(x+y) if 0 < x < < y < 1 and 0 otherwise.
Find the marginal pdf of T if S=X and T = XY. Use the joint pdf of
S = X and T = XY.

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)

19. Let X and Y be continuous random variables with joint pdf:
f(x, y) = x−y for 0 ≤ y ≤ 1 and 1 ≤ x ≤ 2. If U = XY and V = X/Y ,
calculate the joint pdf of U and V , fUV (u, v).

Suppose X and Y are continuous random variables with joint
pdf
f(x,y) = x + y, 0 < x< 1, 0 < y< 1. Let W =
max(X,Y). Find EW.

Let X and Y be two random variables having the joint probability
density fxy= 24xy for 0<x<1, 0<y<1,
0<x+y<1, o elsewhere. Find the joint probability density of Z
= X + Y, and W = 2Y

Suppose the continuous random variables X and Y have joint pdf:
fXY (x, y) = （1/2）xy for 0 < x < 2 and x < y < 2 (a)
Find P(Y < 2X) by integrating in the x direction first. Be
careful setting up your limits of integration. (b) Find P(Y <
2X) by integrating in the y direction first. Be extra careful
setting up your limits of integration. (c) Find the conditional pdf
of X given Y = y,...

Let X and Y are two continuous random variables. It's joint
p.d.f is given as:
f(x,y) = 2 , 0 < x < y < 1
= 0, otherwise
Calculate P(x+y >1)

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