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.
Consider the sample x1, x2, ...,
xn with sample mean x̅ and sample standard deviation s...
Consider the sample x1, x2, ...,
xn with sample mean x̅ and sample standard deviation s
.Let Zi = (xi - x̅ )/s, i = 1,2, ..., n. What
are the values of the sample mean and sample standard deviation of
zi ? Explain the answers with equations.
Suppose X2 and X2 are iid as
Unif([2,6]).
a) What is the cumulative distribution function of...
Suppose X2 and X2 are iid as
Unif([2,6]).
a) What is the cumulative distribution function of
max(X1,X2)?
b) What is the cumulative distribution function of
min(X1,X2)?
Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ...
Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...
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...
Suppose that X1,..., Xn∼iid N(μ,σ2).
a) Suppose that μ is known. What is the MLE of...
Suppose that X1,..., Xn∼iid N(μ,σ2).
a) Suppose that μ is known. What is the MLE of σ?
(b) Suppose that σ is known. What is the MLE of μ?
(c) Suppose that σ is known, and μ has a prior distribution that
is normal with known mean and variance μ0 and
σ02. Find the posterior distribution of μ
given the data.
X1,
X2,...Xn are iid random variables from a U(0,b) distribution. Which
estimator is an unbiased estimator...
X1,
X2,...Xn are iid random variables from a U(0,b) distribution. Which
estimator is an unbiased estimator for b?
2 X bar n
X bar n
1/n (X1squared + X2squared +....Xnsquared)
1/n2(X1squared + X2squared +....Xnsquared)
Consider the following program
Min Z=-x1-x2
s.t
2x1+x2≤10
-x1+2x2≤10
X1, x2≥0
Suppose that the vector c=...
Consider the following program
Min Z=-x1-x2
s.t
2x1+x2≤10
-x1+2x2≤10
X1, x2≥0
Suppose that the vector c= {-1,-1} is
replaced by (-1,-1) +ʎ (2, 3) where ʎ is a real number
Find optimal solutions for all values
of ʎ
Z
X1
X2
S1
S2
RhS
Z
1
0
0
-0.6
-0.2
-8
X1
0
1
0
0.4
-0.2
2
X2...