Question

Given the following joint density,

f(x,y)={10xy^2 if 0<x<y<1

f(x,y)={ 0 otherwise

1. frequency function x given y

2. E(x given y), Var(x given y)

3. Var(E(x given y), E(Var(x given y)

Answer #1

The joint probability density function of x and y is given by
f(x,y)=(x+y)/8 0<x<2, 0<y<2 0 otherwise
calculate the variance of (x+y)/2

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).

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}

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]

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).

If the joint density function of (X,Y) is
f(x,y) =
{ 21/2 * x^2 * y, if x^2 ≤ y ≤ 1
0, otherwise}
What is the correlation coefficient between X and Y? Hint:
calculate the marginal density of X and Y and then EXY, EX, and
EY.

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...

Consider the joint density function f (x, y) = 1 if 0<=
x<= 1; 0<=y<= 1. [0 elsewhere]
a) Obtain the probability density function of the v.a Z, where Z =
X^2.
b) Obtain the probability density function of v.a W, where W =
X*Y^2.
c) Obtain the joint density function of Z and W, that is, g (Z,
W)

Consider the joint pdf
f(x, y) = 3(x^2+ y)/11
for 0 ≤ x ≤ 2, 0 ≤ y ≤ 1.
(a) Calculate E(X), E(Y ),
E(X^2), E(Y^2), E(XY
), Var(X), Var(Y ),
Cov(X, Y ).
(b) Find the best linear predictor of Y given
X.
(c) Plot the CEF and BLP as a function of X.

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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