Question

Let X and Y be random variables, P(X = −1) = P(X = 0) = P(X = 1) = 1/3 and Y take the value 1 if X = 0 and 0 otherwise. Find the covariance and check if random variables are independent.

How to check if they are independent since it does not mean that if the covariance is zero then the variables must be independent.

Answer #1

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 )

Consider the following joint distribution between random
variables X and Y:
Y=0
Y=1
Y=2
X=0
P(X=0, Y=0) = 5/20
P(X=0, Y=1) =3/20
P(X=0, Y=2) = 1/20
X=1
P(X=1, Y=0) = 3/20
P(X=1, Y=1) = 4/20
P(X=1, Y=2) = 4/20
Further, E[X] = 0.55, E[Y] = 0.85, Var[X] = 0.2475 and Var[Y] =
0.6275.
a. (6 points) Find the covariance between X and Y.
b. (6 points) Find E[X | Y = 0].
c. (6 points) Are X and Y 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

Let X and Y be two continuous random variables with joint
probability density function f(x,y) = xe^−x(y+1), 0 , 0< x <
∞,0 < y < ∞ otherwise
(a) Are X and Y independent or not? Why?
(b) Find the conditional density function of Y given X = 1.(

STAT 180 Let X and Y be independent exponential random variables
with mean equals to 4.
1) What is the covariance between XY and X.
2) Let Z = max ( X, Y). Find the Probability Density Function
(PDF) of Z.
3) Use the answer in part 2 to compute the E(Z).

Let X and Y be independent Geometric(p) random variables. What
is P(X<Y)?

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...

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]

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 independent random variables each having the
uniform distribution on [0, 1].
(1)Find the conditional densities of X and Y given that X > Y
.
(2)Find E(X|X>Y) and E(Y|X>Y) .

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