Question

Q1) The joint probability density function of the random variables X and Y is given by ??,? (?, ?) = { ?, 0 < ? < ? < 1 0, ??ℎ?????? a) Find the constant ? b) Find the marginal PDFs of X and Y. c) Find the conditional PDF of X given Y, i.e., ?(?|?) d) Find the variance of X given Y, i.e., ???(?|?) e) Are X and Y statistically independent? Explain why.

Answer #1

4. Let X and Y be random variables having joint probability
density function (pdf) f(x, y) = 4/7 (xy − y), 4 < x < 5 and
0 < y < 1
(a) Find the marginal density fY (y).
(b) Show that the marginal density, fY (y), integrates to 1
(i.e., it is a density.)
(c) Find fX|Y (x|y), the conditional density of X given Y =
y.
(d) Show that fX|Y (x|y) is actually a pdf (i.e., it integrates...

Consider the random variables X and Y with the following joint
probability density function:
fX,Y (x, y) = xe-xe-y, x > 0, y
> 0
(a) Suppose that U = X + Y and V = Y/X. Express X and Y in terms of
U and V .
(b) Find the joint PDF of U and V .
(c) Find and identify the marginal PDF of U
(d) Find the marginal PDF of V
(e) Are U and V independent?

For continuous random variables X and Y with joint probability
density function. f(x,y) = xe−(x+y) when x > 0 and y
> 0 f(x,y) = 0 otherwise
a. Find the conditional density F xly (xly)
b. Find the marginal probability density function fX (x)
c. Find the marginal probability density function fY (y).
d. Explain if X and Y are independent

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

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.(

The joint probability density function (pdf) describing
proportions X and Y of two components in a chemical blend are given
by f(x, y) = 2, 0 < y < x ≤ 1.
(a) Find the marginal pdfs of X and Y.
(b) Find the probability that the combined proportion of these
two components is less than 0.5.
(c) Find the conditional probability density function of Y given
X = x. (d) Find E(Y | X = 0.8).

The joint probability density function of two random variables
(X and Y) is given by fX,Y (x, y) = ( C √y (y ^(α+1)) exp {( −
y(2β+x ^2 ) )/2 } , x ∈ (−∞,∞), y ∈ [0,∞), 0 otherwise. (a) Find C.
(b) Find the marginal density of Y . What type of distribution does
Y follow? (c) Find the conditional density of X | Y . What type of
distribution is this?

Suppose that the joint probability density function of the
random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤
x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.
(a) Sketch the region of non-zero probability density and show
that c = 3/ 2 .
(b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1).
(c) Compute the marginal density function of X and Y...

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 .

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).

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