Question

The joint probability density function (pdf) of X and Y is given by

f(x, y) = cx^2 (1 − y), 0 < x ≤ 1, 0 < y ≤ 1, x + y ≤ 1.

(a) Find the constant c.

(b) Calculate P(X ≤ 0.5).

(c) Calculate P(X ≤ Y)

Answer #1

The solution to this problem is given below

The joint probability density function (pdf) describing
proportions X and Y of two components in a chemical blend are given
by f(x, y) = 2, 0 < y < x ≤ 1.
(a) Find the marginal pdfs of X and Y.
(b) Find the probability that the combined proportion of these
two components is less than 0.5.
(c) Find the conditional probability density function of Y given
X = x. (d) Find E(Y | X = 0.8).

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).

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 .

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...

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...

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).

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).

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 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

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