Question

2. Random variables X and Y have a joint PDF fX,Y (x, y) = 2 for 0 ≤ y ≤ x ≤ 1. Determine (a) E[X] and Var[X]. (b) E[Y ] and Var[Y ]. (c) Cov(X, Y ). (d) E[X + Y ]. (e) Var[X + Y ].

Answer #1

Random Variables X and Y have joint PDF
fX,Y(x,y) =
c*(x+y) , 0<x , x>y
0 ,
otherwise
a. Find the value of the constant c.
b. Find P[x < 1 and y < 2]

Suppose that X and Y are two jointly continuous random variables
with joint PDF
??,(?, ?) =
??
??? 0 ≤ ? ≤ 1 ??? 0 ≤ ? ≤ √?
0
??ℎ??????
Compute and plot ??(?) and ??(?)
Are X and Y independent?
Compute and plot ??(?) and ???(?)
Compute E(X), Var(X), E(Y), Var(Y), Cov(X,Y), and
Cor.(X,Y)

Let X and Y be random variables with the joint pdf
fX,Y(x,y) = 6x, 0 ≤ y ≤ 1−x, 0 ≤ x ≤1.
1. Are X and Y independent? Explain with a picture.
2. Find the marginal pdf fX(x).
3. Find P( Y < 1/8 | X = 1/2 )

* 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).

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 & 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

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?

Suppose X and Y are continuous random variables with joint
density function fX;Y (x; y) = x + y on the square [0; 3] x [0; 3].
Compute E[X], E[Y], E[X2 + Y2], and Cov(3X -
4; 2Y +3).

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(Y < 2X) by integrating in the x direction first. Be
careful setting up your limits of integration. (b) Find P(Y <
2X) by integrating in the y direction first. Be extra careful
setting up your limits of integration. (c) Find the conditional pdf
of X given Y = y,...

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...

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