Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1
0, otherwise. a)...
Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1
0, otherwise. a) Are Y1 and Y2 independent? Why? b) Find Cov(Y1,
Y2). c) Find V(Y1−Y2). d) Find Var(Y1|Y2=y2).
Suppose that Y1 and Y2 are jointly distributed with joint
pdf:
f(y1,y2) ={cy2. for 0 ≤ y1 ≤ y2 ≤ 1
0. otherwise}
(1) Find c value.
(2) Are Y1 and Y2 independent? Justify your answer.
(3) ComputeCov(Y1,Y2).
(4) Find the conditional density of Y1 given Y2=y2.
(5) Using (d), find the conditional expectation E (Y1|Y2).
(6) Suppose that X1=Y1+Y2, and X2= 2Y2. What is the joint
density ofX1andX2?
Suppose that Y1 and Y2 are random variables with joint pdf given
by f(y1,y2) = ky1y2 ; 0 < y1 <y2 <1,
where k is a constant equal to 8.
a) Find the conditional expected value and variance of Y1 given
Y2=y2.
b) Are Y1 and Y2 independent? Justify your answer.
c) Find the covariance and correlation between Y1 and Y2.
d) Find the expected value and variance of Y1+Y2.
Let (A,B) have joint PDF f(a,b)=(ca^2b^2 when 0 < a,b,a+b
< 1 and 0 otherwise for some constant c > 0. 1. Find a
formula for E[A | B = b]. 2. Find Cov(A,B).
Let ?1 and ?2 have the joint pdf
f (?1, ?2)= 6?2 0<?2<?1<1
=0 else where
A. Find conditional mean and conditional variance ?1given?2
.
B. Theorem of total mean and total variance?1given?2
.(urgently needed)
The joint density of Y1, Y2 is given by f(y) = k, −1 ≤ y1 ≤ 1, 0
≤ y2 ≤ 1, y1 + y2 ≤ 1, y1 − y2 ≥ −1, 0, otherwise
a. Find the value of k that makes this a probability density
function.
b. Find the probabilities P(Y2 ≤ 1/2) and P(Y1 ≥ −1/2, Y2 ≤
1/2
c. Find the marginal distributions of Y1 and of Y2.
d. Determine if Y1 and Y2 are independent
e....
Let X and Y have the joint pdf f(x, y) = 8xy, 0 ≤ x ≤ y ≤ 1. (i)
Find the conditional means of X given Y, and Y given X. (ii) Find
the conditional variance of X given Y. (iii) Find the correlation
coefficient between X and Y.
Let X1 and X2 have the joint pdf
f(x1,x2) = 2 0<x1<x2<1; 0. elsewhere
(a) Find the conditional densities (pdf) of X1|X2 = x2 and X2|X1
= x1.
(b) Find the conditional expectation and variance of X1|X2 = x2 and
X2|X1 = x1.
(c) Compare the probabilities P(0 < X1 < 1/2|X2 = 3/4) and
P(0 < X1 < 1/2).
(d) Suppose that Y = E(X2|X1). Verify that E(Y ) = E(X2), and that
var(Y ) ≤ var(X2).
Let c(y1, y2) = y1 + y2 + (y1y2)^ −(1/3). Does this cost
function have economies of scale for y1? What about economies of
scope for any strictly positive y1 and y2. Hint, economies of scope
exist if for a positive set of y1 and y2, c(y1, y2) < c(y1, 0) +
c(0, y2). [Hint: Be very careful to handle the case of y2 = 0
separately.]
Let y1, y2, .....y10 be a random sample from an exponential pdf
with unknown parameter λ. Find the form of the GLRT for H0: λ=λ0
versus H1:λ ≠ λ0. What integral would have to be evaluated to
determine the critical value if α were equal to 0.05?