Question

A two dimensional variable (X, Y) is uniformly distributed over
the square having the vertices (2, 0), (0, 2), (-2, 0), and (0,
-2).

(i) Find the joint probability density function.

(ii) Find associated marginal and conditional distributions.

(iii) Find E(X), E(Y), Var(X), Var(Y).

(iv) Find E(Y|X) and E(X|Y).

(v) Calculate coefficient of correlation between X and Y.

Answer #1

X is uniformly distributed on the interval (0, 1), and Y is
uniformly distributed on the interval (0, 2). X and Y are
independent. U = XY and V = X/Y .
Find the joint and marginal densities for U and V .

. (X,Y ) is uniformly distributed over the triangle with
vertices (0,0),(2,0),(0,1). Find the density f(z) of X −Y for z ≤
0.

Suppose that X is a random variable uniformly distributed over
the interval (0, 2), and Y is a random variable uniformly
distributed over the interval (0, 3). Find the probability density
function for X + Y .

Suppose that X and Y are continuous and jointly distributed by
f(x, y) = c(x + y)2 on the triangular region defined by
0 ≤ y ≤ x ≤ 1.
a. Find c so that we have a joint pdf.
b. Find the marginal for X
c. Find the marginal for Y.
d. Find E[X] and V[X].
e. Find E[Y] and V[Y].
f. Find E[XY]
g. Find cov(X, Y).
h. Find the correlation coefficient for the two variables.
i. Prove...

1. Let P = (X, Y ) be a uniformly distributed random point over
the diamond with vertices (1, 0),(0, 1),(?1, 0),(0, ?1). Show that
X and Y are not independent but E[XY ] = E[X]E[Y ]

If the joint density function of (X,Y) is
f(x,y) =
{ 21/2 * x^2 * y, if x^2 ≤ y ≤ 1
0, otherwise}
What is the correlation coefficient between X and Y? Hint:
calculate the marginal density of X and Y and then EXY, EX, and
EY.

Let the random vector (X, Y ) be drawn uniformly from the disk D
= {(x, y) : x2 +y2 ≤ 1}.
(a) (2 pts) What is the joint density function of (X, Y )?
(b) (4 pts) What is the marginal density function of Y ?
(c) (4 pts) What is the conditional density of X given Y =
1/2?

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 the joint p.d.f. f(x, y) = 1I(|x^2−y^2|<1).
Then,
(a) Find the marginal distributions of X and Y respectively.
(b) Obtain the conditional distribution of Y given X=x,for 0<
x <1.
(c) Find the mean and variance of X only.

the
random variable c is uniformly distributed over the integral (5,10)
a) if y=G(x)=4(x^2) , find fy(y)
b) determine E{G(x)}

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