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 ).

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 .

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).

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}

X and Y are continuous random variables. Their joint probability
density function is given as f(x,y) = 1/5 (y+2) for 0<y<1 and
y-1<x<y+1. Calculate the conditional expectation
E(x/y=0).
Please show all the work and explain if the answer will be a
number or just y in a given range.

X and Y are continuous random variables. Their joint probability
distribution function is :
f(x,y) = 1/5(y+2) , 0 < y < 1, y-1 < x < y +1
= 0, otherwise
a) Find marginal density of Y, fy(y)
b) Calculate E[X | Y = 0]

A joint density function of the continuous random variables
x and y is a function f(x,
y) satisfying the following properties.
f(x, y) ≥ 0 for all (x, y)
∞
−∞
∞
f(x, y) dA = 1
−∞
P[(x, y) R] =
R
f(x, y) dA
Show that the function is a joint density function and find the
required probability.
f(x, y) =
1
8
,
0 ≤ x ≤ 1, 1 ≤ y ≤ 9
0,
elsewhere
P(0 ≤...

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