Question

If f(x,y) = k is a joint probability density function over the region 0<x<4, 0<y, and x-1<y<x+1, what is the value of f(x)?

Answer #1

f (x, y) =(x+y)/(5(5 + 2)) is a joint probability density
function over the range 0 < x < 5 and 0 < y < 2. Find V
(X). Please report your answer to 3 decimal places.

STAT 190 Let X and Y have the joint probability density function
(PDF), f X,Y (x, y) = kx, 0 < x < 1, 0 < y < 1 -
x^2,
= 0, elsewhere,
where k is a constant.
1) What is the value of k.
2)What is the marginal PDF of X.
3) What is the E(X^2 Y).

Let X and Y have the joint probability density function f(x, y)
= ⎧⎪⎪ ⎨ ⎪⎪⎩ ke−y , if 0 ≤ x ≤ y < ∞, 0, otherwise. (a) (6pts)
Find k so that f(x, y) is a valid joint p.d.f. (b) (6pts) Find the
marginal p.d.f. fX(x) and fY (y). Are X and Y independent?

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)

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) 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 .

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
)?

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

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 ≤...

Problem 4 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.
Problem 5 Find the expected values E
(X) and E (Y) for the density function given
in Problem 4.
Problem 7. Using information from problems 4
and 5, find
Cov(X,Y).

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