Let the random variable and have the joint pmf X Y f(x,y) =
{x(y)^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 . σ2 X
(e) Find . σ2 Y
(f) Find Cov . (X,Y )
(g) Find p , Corr . (x,y)
(h) Are and X and Y independent
let the continuous random variables X and Y have the joint
pdf:
f(x,y)=6x , 0<x<y<1
i) find the marginal pdf of X and Y respectively,
ii) the conditional pdf of Y given x, that is
fY|X(y|x),
iii) E(Y|x) and Corr(X,Y).
Let X and Y be continuous random variables with joint pdf
f(x, y) = kxy^2 0 < x, 0 < y, x
+ y < 2
and 0 otherwise
1) Find P[X ≥ 1|Y ≤ 1.5]
2) Find P[X ≥ 0.5|Y ≤ 1]
Let the random variable X and Y have the joint pmf f(x, y) = , c
xy 2 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 μ . X
(c) Find μ . Y
(d) Find σ . 2 X
(e) Find σ . 2 Y
(f) Find Cov (X, Y )...
Let X and Y be two jointly continuous random variables with
joint PDF
f(x,y) = Mxy^2
0<x<y<1
a) Find M = ?
b) Find the marginal probability densities.
c) P( y> 1/2 | x = .25) = ?
d) Corr (x,y) = ?
. Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { 1, 0 < x, y < 1 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
..
The joint density function for random variables X,
Y, and Z is
f(x, y,
z)= Cxyz if 0 ≤
x ≤ 1, 0 ≤ y ≤ 2, 0 ≤
z ≤ 2, and
f(x, y,
z) = 0 otherwise.
(a) Find the value of the constant C.
(b) Find P(X ≤ 1, Y ≤ 1, Z ≤ 1).
(c) Find P(X + Y + Z ≤ 1).
Suppose X and Y are random variables with joint density f(x, y)
= c(x2y + y2), − 1 ≤ x ≤ 1, 0 ≤ y ≤ 1 (0
else).
a) Find c.
b) Determine whether X and Y are independent.
c) Compute P(3X + 2Y > 1 | −1/2 ≤ X ≤ 1/2).
Suppose that we have two random variables (Y,X) with joint
probability density function f(y,x). Consider the following
estimator of the intercept of the Best Linear Predictor:
A = ?̅ - B • ?̅ ,
where ?̅ is the sample mean of y, ?̅ is the sample mean of x, and B
is the sample covariance of Y and X divided by the sample variance
of X.
Identify the probability limit of A (if any). For each step in
your derivation,...
A joint pdf is defined as f(x) =cxy for x in [1,2] and y in
[4,5]
(a) What is the value of the constant c?
(b) Are X and Y independent? Explain.
(c) What is the covariance oc X and Y? i.e. Cov(X ,Y)