Question

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 and hence calculate E [X] and E [Y ].

(d) Find the conditional density function of X given Y = y.

(e) Determine the covariance between X and Y , cov [X, Y ]=E[XY ]E [X]E[Y ].

(f) State, giving reasons, whether X and Y are independent. (g) Find V ar[XY ]

Answer #1

The Joint density function of the random variables X and Y is

a) The region is shown below

Now as total probability is 1, so

b)

As , X and Y are continuous random variables so their sum is also continuous .

Hence,

c) The marginal PDF's are given by

and

Hence,

and

d) Using definition of conditional PDF we get

**NOTE: As per
rules, only 4 parts will be solved at a time**

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.(

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 ).

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

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

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 .

The random variables X and Y have a joint density function given
by f(x, y) = ( 2e(−2x) /x, 0 ≤ x < ∞, 0 ≤ y ≤ x , otherwise.
(a) Compute Cov(X, Y ).
(b) Find E(Y | X).
(c) Compute Cov(X,E(Y | X)) and show that it is the same as
Cov(X, Y ).
How general do you think is the identity that Cov(X,E(Y |
X))=Cov(X, Y )?

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...

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).

Suppose that X and Y have the following
joint probability density function.
f (x, y) =
3
332
y, 0 < x < 5, y
> 0, x − 4 < y < x +
4
(a)
Find E(XY).
(b)
Find the covariance between X and Y.

2.
The joint probability density function of X and Y is given
by
f(x,y) = (6/7)(x² + xy/2),
0 < x < 1, 0 < y < 2. f(x,y) =0
otherwise
a) Compute the marginal densities of X and Y. b) Are X and Y
independent. c) Compute the conditional density
function f(y|x) and check restrictions on function you derived d)
probability P{X+Y<1}

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