Question

Suppose that X and Y have joint probability density function given by: f(x, y) = 2 for 0 ≤ x ≤ 1 and 0 ≤ y ≤ x. What is Cov(X, Y )?

Answer #1

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 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...

Suppose that X and Y have the following joint probability
density function. f (x, y) = 3 /146 *x, 0 < x < 5, y
> 0, x − 2 < y < x + 2 Find E(X).

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}

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 )?

7. Suppose that random variables X and Y have a joint density
function given by: f(x, y) = ? + ? 0 ≤ ?≤ 1, 0 ≤ ? ≤ 1
(a) Find the density functions of X and Y, f(x) and f(y).
(b) Find E[X] and Var(Y).

Suppose X and Y are continuous random variables with joint
density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
(a). Compute the joint CDF F(x,y).
(b). Compute the marginal density for X and Y .
(c). Compute Cov(X,Y ). Are X and Y independent?

2.
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} [5+5+5+5 = 20]

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).

9. Suppose X and Y are continuous random variables with joint
density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
(a). Compute the joint CDF F(x,y).
(b). Compute the marginal density for X and Y .
(c). Compute Cov(X,Y ). Are X and Y independent?

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