Question

Let X and Y be a random variables with the joint probability density function fX,Y (x, y) = { e −x−y , 0 < x, y < ∞ 0, otherwise } . a. Let W = max(X, Y ) Compute the probability density function of W. b. Let U = min(X, Y ) Compute the probability density function of U. c. Compute the probability density function of X + Y .

Answer #1

Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { cx2y, 0 < x2 < y < x for
x > 0 0, otherwise }. compute the marginal probability density
functions fX(x) and fY (y). Are the random variables X and Y
independent?.

Let fX,Y be the joint density function of the random variables X
and Y which is equal to fX,Y (x, y) = { x + y if 0 < x, y <
1, 0 otherwise. } Compute the probability density function of X + Y
. Referring to the problem above, compute the marginal probability
density functions fX(x) and fY (y). Are the random variables X and
Y independent?

Consider the random variables X and Y with the following joint
probability density function:
fX,Y (x, y) = xe-xe-y, x > 0, y
> 0
(a) Suppose that U = X + Y and V = Y/X. Express X and Y in terms of
U and V .
(b) Find the joint PDF of U and V .
(c) Find and identify the marginal PDF of U
(d) Find the marginal PDF of V
(e) Are U and V independent?

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?

* The random variables X and Y have a joint density function
given by fX,Y(x, y) = ⇢ 1/y, 0 < y < 1, 0 < x < y, 0,
otherwise. Compute (a) Cov(X,Y) and (b) Corr(X,Y).

1. Let (X; Y ) be a continuous random vector with joint
probability density function
fX;Y (x, y) =
k(x + y^2) if 0 < x < 1 and 0 < y < 1
0 otherwise.
Find the following:
I: The expectation of XY , E(XY ).
J: The covariance of X and Y , 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 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.(

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).

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