Question

Suppose X and Y are jointed distributed random variables with joint pdf f(x,y) given by

f(x,y) = 8xy 0<y<x<1

= 0 elsewhere

What is P(0<X<1/2 , 1/4<Y<1/2)

Answer #1

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...

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.

a) The joint probability density function of the random
variables X, Y is given as
f(x,y) =
8xy
if 0≤y≤x≤1 , and 0
elsewhere.
Find the marginal probability density functions.
b) Find the expected values EX and
EY for the density function above
c) find Cov X,Y .

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 random variables with joint pdf f(x, y) = 2 + x −
y, for 0 <= x <= 1, 1 <= y <= 2.
(a) Find the probability that min(X, Y ) <= 1/2.
(b) Find the probability that X + √ Y >= 4/3.

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).

Problem 4 The joint probability density
function of the random variables X, Y is given as
f(x,y)=8xy
if 0 ≤ y ≤ x ≤ 1, and 0 elsewhere.
Find the marginal probability density functions.
Problem 5 Find the expected values E
(X) and E (Y) for the density function given
in Problem 4.
Problem 7. Using information from problems 4
and 5, 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 the random variable (X, Y ) has a joint pdf for the
form
?cxy 0≤x≤1,0≤y≤1 f(x,y) = .
0 elsewhere
(a) (5 pts) Find c so that f is a valid distribution.
(b) (6 pts) Find the marginal distribution, g(x) for X and the
marginal distribution for Y , h(y).
(c) (6 pts) Find P (X > Y ).
(d) (6 pts) Find the pdf of X +Y.
(e) (6 pts) Find P (Y < 1/2|X > 1/2).
(f)...

Let X and Y be random variables with the joint pdf
fX,Y(x,y) = 6x, 0 ≤ y ≤ 1−x, 0 ≤ x ≤1.
1. Are X and Y independent? Explain with a picture.
2. Find the marginal pdf fX(x).
3. Find P( Y < 1/8 | X = 1/2 )

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