Question

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?

Answer #1

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

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.

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

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.

Suppose that X and Y are continuous and jointly distributed by
f(x, y) = c(x + y)2 on the triangular region defined by
0 ≤ y ≤ x ≤ 1.
a. Find c so that we have a joint pdf.
b. Find the marginal for X
c. Find the marginal for Y.
d. Find E[X] and V[X].
e. Find E[Y] and V[Y].
f. Find E[XY]
g. Find cov(X, Y).
h. Find the correlation coefficient for the two variables.
i. Prove...

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.

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

RVs Х and Y are continuous, joint PDF fx,y(x,y) = cxy, if 0 ≤ x
≤ y ≤ 1 , and fx,y(x,y) = 0, otherwise, c is a constant.
Find a) fx|y(x|y = 0.5) for x ∈ [0, 0.5],
b) E(X | y = 0.5).

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