Question

Let X and Y have the joint PDF **(i really just need g and
h if that makes it easier)**

f(x) = { c(y + x^2) 0 < x < 1 and 0 < y < 1 ; 0 elsewhere

a) Find c such that this is a PDF.

b) What is P(X ≤ .4, Y ≤ .2) ?

C) Find the Marginal Distribution of X, f(x)

D) Find the Marginal Distribution of Y, f(y)

E) Are X and Y independent? Explain

F) Find f(x|y)

G) Find Cov(X, Y)

H) Find p

Answer #1

Suppose the random variable (X, Y ) has a joint pdf for the
form
?cxy 0≤x≤1,0≤y≤1 f(x,y) = .
0 elsewhere
(a) (5 pts) Find c so that f is a valid distribution.
(b) (6 pts) Find the marginal distribution, g(x) for X and the
marginal distribution for Y , h(y).
(c) (6 pts) Find P (X > Y ).
(d) (6 pts) Find the pdf of X +Y.
(e) (6 pts) Find P (Y < 1/2|X > 1/2).
(f)...

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).

Let X, Y be two random variables with a joint pmf
f(x,y)=(x+y)/12 x=1,2 and y=1,2
zero elsewhere
a)Are X and Y discrete or continuous random variables?
b)Construct and joint probability distribution table by writing
these probabilities in a rectangular array, recording each marginal
pmf in the "margins"
c)Determine if X and Y are Independent variables
d)Find P(X>Y)
e)Compute E(X), E(Y), E(X^2) and E(XY)
f)Compute var(X)
g) Compute cov(X,Y)

Let X and Y have the joint pdf f(x,y) = 6*(x^2)*y for 0 <= x
<= y and x + y <= 2.
What is the marginal pdf of X and Y?
What is P(Y < 1.1 | X = 0.6)?
Are X and Y dependent random variables?

Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and
0<=y<=1.
a) Find k.
b) Find the joint cumulative density function of (X,Y)
c) Find the marginal pdf of X and Y.
d) Find Pr[Y<X2] and Pr[X+Y>0.5]

Let
X & Y be two continuous random variables with joint pdf:
fXY(X,Y) = { 2 x+y =< 1, x >0, y>0
{ 0 otherwise
find Cov(X,Y) and ρX,Y

Let X and Y be continuous random variables with joint
distribution function F(x, y), and let g(X, Y ) and h(X, Y ) be
functions of X and Y . Prove the following:
(a) E[cg(X, Y )] = cE[g(X, Y )].
(b) E[g(X, Y ) + h(X, Y )] = E[g(X, Y )] + E[h(X, Y )].
(c) V ar(a + X) = V ar(X).
(d) V ar(aX) = a 2V ar(X).
(e) V ar(aX + bY ) = a...

Let the random variable X and Y have the joint pmf f(x, y) =
xy^2/c where x = 1, 2, 3; y = 1, 2, x + y ≤ 4 , that is, (x, y) are
{(1, 1),(1, 2),(2, 1),(2, 2),(3, 1)} .
(a) Find c > 0 .
(b) Find μX
(c) Find μY
(d) Find σ^2 X
(e) Find σ^2 Y
(f) Find Cov (X, Y )
(g) Find ρ , Corr (X, Y )
(h) Are X...

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...

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...

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