Question

Consider continuous random variables X and Y whose joint pdf is f(x, y) = 1 with 0 < y <1 – abs(x). Show that Cov(X, Y ) = 0 even though X and Y are dependent. Note: For this

problem, you only need to show that the covariance is zero. You need not show that X and Y are dependent.

Answer #1

Consider continuous random variables X and Y whose joint pdf is
f(x, y) = 1 with 0 < y < 1 − |x|. Show that Cov(X, Y ) = 0
even though X and Y are dependent. Note: For this problem, you only
need to show that the covariance is zero. You need not show that X
and Y are dependent.

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 & 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

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) = 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)

X and Y are jointly continuous with joint pdf
f(x, y) = 2, x > 0, y > 0, x + y ≤ 1
and 0 otherwise.
a) Find marginal pdf’s of X and of Y.
b) Find covariance Cov(X,Y).
c) Find correlation Corr(X,Y). What you can say about the
relationship between X and Y?

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).

The continuous random variables X and Y have joint pdf f(x, y) =
cy2 + xy/3 0 ≤ x ≤ 2, 0 ≤ y ≤ 1 (a)
What is the value of c that makes this a proper pdf? (b) Find the
marginal distribution of X. (c) (4 points) Find the marginal
distribution of Y . (d) (3 points) Are X and Y independent? Show
your work to support your answer.

Suppose X and Y are continuous random variables with joint
density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
(a). Compute the joint CDF F(x,y).
(b). Compute the marginal density for X and Y .
(c). Compute Cov(X,Y ). Are X and Y independent?

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