Let X1,...,Xn be iid from Poisson(theta), Set
k(theta)=exp(-theta).
(a) What is the MLE of k(theta)? Is...
Let X1,...,Xn be iid from Poisson(theta), Set
k(theta)=exp(-theta).
(a) What is the MLE of k(theta)? Is it unbiased?
(b) Obtain the CRLB for any unbiased estimator of k(theta).
Find the number of integer solutions to x1+x2+x3=20 given the
following restrictions:
(A) x1>=3, x2>=2,x3>=5
(B)...
Find the number of integer solutions to x1+x2+x3=20 given the
following restrictions:
(A) x1>=3, x2>=2,x3>=5
(B) x1>=0, x2>=0, x3<=6
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) =...
Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then,
Show the density of the statistic T = X(n) is given by
FX(n) (x) = n/ab * (x/a)^{n/(b-1}} for 0 <= x <=
a ; otherwise zero.
# using the following
P (X(n) < x ) = P (X1 < x, X2 < x, ,,,,,,,,, Xn < x
),
Then assume...
Find the number of solutions to
x1+x2+x3+x4=16 with
integers x1 ,x2, x3, x4
satisfying
(a) xj ≥...
Find the number of solutions to
x1+x2+x3+x4=16 with
integers x1 ,x2, x3, x4
satisfying
(a) xj ≥ 0, j = 1, 2, 3, 4;
(b) x1 ≥ 2, x2 ≥ 3, x3 ≥ −3,
and x4 ≥ 1;
(c) 0 ≤ xj ≤ 6, j = 1, 2, 3, 4
Suppose that X1, X2, X3,
X4 are iid N(θ,4). We wish to test H0: θ =...
Suppose that X1, X2, X3,
X4 are iid N(θ,4). We wish to test H0: θ = 2
vs H1: θ = 5. Consider the following tests:
Test 1: Reject H0 iff X1 > 4.7
Test 1: Reject H0 iff 1/3(X1 +
2X2) > 4.5
Test 3: Reject H0 iff 1/2(X1 +
X3) > 4.2
Test 4: Reject H0 iff x̄>4.1 (xbar > 4.1)
Find Type 1 and Type 2 error probabilities for each test and
compare the tests.
6. Let X1, X2, ..., Xn be a random sample of a random variable X
from...
6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...
Choose the all correct things
Sn=X1+X2+...+Xn
1. E[Sn]=2n, when RV X1,...,Xn is iid and E[X]=2
2....
Choose the all correct things
Sn=X1+X2+...+Xn
1. E[Sn]=2n, when RV X1,...,Xn is iid and E[X]=2
2. VAR[Sn]=3n, when RV X1,...,Xn is iid and VAR[X]=3
3. VAR[Sn]=2n is always correct, when RV X1,...,Xn is
identically distributed and VAR[X]=2
4 .Sn is an Exponential distribution with a mean of 2n and a
variance of 3n ,when RV X1,...,Xn is iid and E[X]=2, VAR[X]=3
Exponential distribution
A random sample X1, X2, . . . , Xn is drawn from a population
with...
A random sample X1, X2, . . . , Xn is drawn from a population
with pdf. f(x; β) = (3x^2)/(β^3) , 0 ≤ x ≤ β 0, otherwise
(a) [6] Find the pdf of Yn, the nth order statistic of the
sample.
(b) [4] Find E[Yn].
(c) [4] Find Var[Yn].
(d)[3] Find the mean squared error of Yn when Yn is used as a
point estimator for β
(e) [2] Find an unbiased estimator for β.
2. Find the number of integer solutions to x1 + x2 + x3 + x4 +...
2. Find the number of integer solutions to x1 + x2 + x3 + x4 +
x5 = 50, x1 ≥ −3, x2 ≥ 0, x3 ≥ 4, x4 ≥ 2, x5 ≥ 12.
Consider the following linear program Max 5x1+5x2+3x3
St
x1+3x2+x3<=3
-x1+ 3x3<=2
2x1-x2 +2x3<=4
2x1+3x2-x3<=2
xi>=0 for...
Consider the following linear program Max 5x1+5x2+3x3
St
x1+3x2+x3<=3
-x1+ 3x3<=2
2x1-x2 +2x3<=4
2x1+3x2-x3<=2
xi>=0 for i=1,2,3
Suppose that while solving this problem with Simplex method, you
arrive at the following table:
z
x1
x2
x3
x4
x5
x6
x7
rhs
Row0
1
0
-29/6
0
0
0
11/6
2/3
26/3
Row1
0
0
-4/3
1
0
0
1/3
-1/3
2/3
Row2
0
1
5/6
0
0
0
1/6
1/3
4/3
Row3
0
0
7/2
0
1
0
-1/2
0...