Question

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?

Answer #1

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?

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

Suppose X and Y are continuous random variables with joint
density function fX;Y (x; y) = x + y on the square [0; 3] x [0; 3].
Compute E[X], E[Y], E[X2 + Y2], and Cov(3X -
4; 2Y +3).

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 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 ≤...

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 .

Let X and Y be jointly continuous random variables with joint
density function f(x, y) = c(y^2 − x^2 )e^(−2y) , −y ≤ x ≤ y, 0
< y < ∞.
(a) Find c so that f is a density function.
(b) Find the marginal densities of X and Y .
(c) Find the expected value of X

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 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 )?

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