Question

Let continuous random variables X, Y be jointly continuous, with
the following joint distribution f_{XY}(x,y) =
e^{-x-y} for x≥0, y≥0 and f_{XY}(x,y) = 0
otherwise.

1) Sketch the area where f_{XY}(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.

Answer #1

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

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 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, 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 two continuous random variables with joint
probability density function f(x,y) = xe^−x(y+1), 0 , 0< x <
∞,0 < y < ∞ otherwise
(a) Are X and Y independent or not? Why?
(b) Find the conditional density function of Y given X = 1.(

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(X < 1, Y < 1). (b) Use the joint pdf to find P(Y >
1). Be careful setting up your limits of integration. (c) Find the
marginal pdf of Y , fY (y). Be sure to state the support. (d) Use
the marginal pdf of Y to find P(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.

Let X and Y be two continuous random variables with joint
probability density function
?(?, ?) = { ? 2 + ?? 3 0 ≤ ? ≤ 1, 0 ≤ ? ≤ 2 0 ??ℎ??????
Find ?(? + ? ≥ 1). Sketch the surface in the ? − ? plane.

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 be two continuous random variables with joint
probability density function
f(x,y) =
6x 0<y<1, 0<x<y,
0 otherwise.
a) Find the marginal density of Y .
b) Are X and Y independent?
c) Find the conditional density of X given Y = 1 /2

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