Question

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).

Answer #1

Q1) The joint probability density function of the random
variables X and Y is given by ??,? (?, ?) = { ?, 0 < ? < ?
< 1 0, ??ℎ?????? a) Find the constant ? b) Find the marginal
PDFs of X and Y. c) Find the conditional PDF of X given Y, i.e.,
?(?|?) d) Find the variance of X given Y, i.e., ???(?|?) e) Are X
and Y statistically independent? Explain why.

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)

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 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

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.(

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}

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).

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?

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 )

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