Question

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 to 1.) Find IE(Y ).

(e) State, but do not evaluate, the integral whose value is IE(e^XY ).

(f) Are X and Y independent? Explain. (You do not need to do any additional calculations to answer this question.)

Answer #1

1. Let (X,Y ) be a pair of random variables with joint pdf given
by f(x,y) = 1(0 < x < 1,0 < y < 1).
(a) Find P(X + Y ≤ 1).
(b) Find P(|X −Y|≤ 1/2).
(c) Find the joint cdf F(x,y) of (X,Y ) for all (x,y) ∈R×R.
(d) Find the marginal pdf fX of X. (e) Find the marginal pdf fY
of Y .
(f) Find the conditional pdf f(x|y) of X|Y = y for 0...

Let X and Y be continuous random variables with joint density
function f(x,y) and marginal density functions fX(x) and fY(y)
respectively. Further, the support for both of these marginal
density functions is the interval (0,1).
Which of the following statements is always true? (Note there
may be more than one)
E[X^2Y^3]=(∫0 TO 1 x^2 dx)(∫0 TO 1 y^3dy)
E[X^2Y^3]=∫0 TO 1∫0 TO 1x^2y^3 f(x,y) dy dx
E[Y^3]=∫0 TO 1 y^3 fX(x) dx
E[XY]=(∫0 TO 1 x fX(x)...

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 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...

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?

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

Suppose the continuous random variables X and Y have joint pdf:
fXY (x, y) = （1/2）xy for 0 < x < 2 and x < y < 2 (a)
Find P(X < 1, Y < 1). (b) Use the joint pdf to find P(Y >
1). Be careful setting up your limits of integration. (c) Find the
marginal pdf of Y , fY (y). Be sure to state the support. (d) Use
the marginal pdf of Y to find P(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?

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