Question

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?

Answer #1

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 X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { e −x−y , 0 < x, y < ∞ 0,
otherwise } . a. Let W = max(X, Y ) Compute the probability density
function of W. b. Let U = min(X, Y ) Compute the probability
density function of U. c. Compute the probability density function
of X + Y .

A joint density function is given by fX,Y (x, y) = ( kx, 0 <
x < 1, 0 < y < 1 0, otherwise.
(a) Calculate k
(b) Calculate marginal density function fX(x)
(c) Calculate marginal density function fY (y)
(d) Compute P(X < 0.5, Y < 0.1)
(e) Compute P(X < Y )
(f) Compute P(X < Y |X < 0.5)
(g) Are X and Y independent random variables? Show your
reasoning (no credit for yes/no answer).
(h)...

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 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)...

Let X and Y be independent random variables with density functions given by fX (x) = 1/2, −1 ≤ x ≤ 1 and fY (y) = 1/2, 3 ≤ y ≤ 5. Find the density function of X-Y.

* The random variables X and Y have a joint density function
given by fX,Y(x, y) = ⇢ 1/y, 0 < y < 1, 0 < x < y, 0,
otherwise. Compute (a) Cov(X,Y) and (b) Corr(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

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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