Question

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).

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 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.

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?

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.

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.

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.

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.

1. Let (X,Y ) be a pair of random variables with joint pdf given
by f(x,y) = 1(0 < x < 1,0 < y < 1).
(a) Find P(X + Y ≤ 1).
(b) Find P(|X −Y|≤ 1/2).
(c) Find the joint cdf F(x,y) of (X,Y ) for all (x,y) ∈R×R.
(d) Find the marginal pdf fX of X. (e) Find the marginal pdf fY
of Y .
(f) Find the conditional pdf f(x|y) of X|Y = y for 0...

1. Let (X; Y ) be a continuous random vector with joint
probability density function
fX;Y (x, y) =
k(x + y^2) if 0 < x < 1 and 0 < y < 1
0 otherwise.
Find the following:
I: The expectation of XY , E(XY ).
J: The covariance of X and Y , Cov(X; Y ).

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