Question

Let X and Y have a joint density function given by f(x; y) = 3x; 0 <= y <= x <= 1

(a) Find P(X<2Y).

(b) Find cov(X,Y).

(c) Find P(X < 1/2 |Y = 1/3).

(d) Find P(X = 1/2|Y = 1/3).

(e) Find P(X > 1/2|Y > 1/3).

(f) Find the conditional expectation E(X|Y = y).

Answer #1

Let X and Y have joint density f given by f(x, y) = cxy 0 ≤ y ≤
x, 0 ≤ x ≤ 1.
(a) Determine the normalization constant c.
(b) Determine P(X + 2Y ≤ 1).
(c) Find E(X|Y = y).
(d) Find E(X).

Suppose that the joint density function of X and
Y is given by
f (x, y) =
45 xe−3x(y +
5) x > 0,
y > 0.
(a)
Find the conditional density of X, given Y
= y.
(b)
Find the conditional density of Y, given X
= x.
(c)
Find P(Y > 5 | X = 4).

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...

The joint density function of (X, Y ) is f(x, y) = c(x + y), 0 ≤
y ≤ x ≤ 1.
(1) Find c.
(2) Find the conditional density f(y|x).
(3) Find P(Y > 0.3|X = 0.5).

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 )?

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 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

A joint density function is given by fX,Y (x, y) = ( kx, 0 <
x < 1, 0 < y < 1 0, otherwise.
(a) Calculate k
(b) Calculate marginal density function fX(x)
(c) Calculate marginal density function fY (y)
(d) Compute P(X < 0.5, Y < 0.1)
(e) Compute P(X < Y )
(f) Compute P(X < Y |X < 0.5)
(g) Are X and Y independent random variables? Show your
reasoning (no credit for yes/no answer).
(h)...

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.

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
)?

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