Question

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]

Answer #1

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}

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...

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

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)...

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.(

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...

Let X and Y have joint density f(x, y) = 6/7(x + y)^2 if 0 ≤ x ≤
1, 0 ≤ y ≤ 1, 0 otherwise, where c is a positive constant.
Compute the marginal densities of X and of Y (be explicit about
all cases!).
Compute P(Y + 2X < 1).
Determine whether X and Y are independent. Justify your
answer.

Given the joint probability density function f ( x , y ) for 0
< x < 3 and 0 < y < 2 x^2y/81 Find the conditional
probability distribution of X=1 given that Y = 1 f ( x , y ) = x^2
y/ 81 . F i n d the conditional probability distribution of X=1
given that Y = 1. i . e . f (X ∣ y = 1 )( 1 )

The joint probability density function of two random variables
(X and Y) is given by fX,Y (x, y) = ( C √y (y ^(α+1)) exp {( −
y(2β+x ^2 ) )/2 } , x ∈ (−∞,∞), y ∈ [0,∞), 0 otherwise. (a) Find C.
(b) Find the marginal density of Y . What type of distribution does
Y follow? (c) Find the conditional density of X | Y . What type of
distribution is this?

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