For X1, ..., Xn iid Unif(0, 1):
a) ShowX(j) ∼Beta(j,n+1−j)
b)Find the joint pdf between X(1)...
For X1, ..., Xn iid Unif(0, 1):
a) ShowX(j) ∼Beta(j,n+1−j)
b)Find the joint pdf between X(1) and X(n)
c) Show the conditional pdf X(1)|X(n) ∼ X(n)Beta(1, n − 1
Let X1,…, Xn be a sample of iid random
variables with pdf f (x; ?) =...
Let X1,…, Xn be a sample of iid random
variables with pdf f (x; ?) = 3x2 /(?3) on S
= (0, ?) with Θ = ℝ+. Determine
i) a sufficient statistic for ?.
ii) F(x).
iii) f(n)(x)
Suppose X1,..., Xn are iid with pdf f(x;θ) = 2x / θ2,
0 < x ≤...
Suppose X1,..., Xn are iid with pdf f(x;θ) = 2x / θ2,
0 < x ≤ θ. Find I(θ) and the Cramér-Rao lower bound for the
variance of an unbiased estimator for θ.
Let X1, X2, …, Xn be iid with pdf ?(?|?) = ? −(?−?)? −? −(?−?) ,...
Let X1, X2, …, Xn be iid with pdf ?(?|?) = ? −(?−?)? −? −(?−?) ,
−∞ < ? < ∞. Find a C.S.S of θ
Let X1,…, Xn be a sample of iid random variables with
pdf f (x ∶ ?)...
Let X1,…, Xn be a sample of iid random variables with
pdf f (x ∶ ?) = 1/? for x ∈ {1, 2,…, ?} and Θ = ℕ. Determine the
MLE of ?.
Let X i ~ Unif(0, 1) for 1 <= i <= n be IID (independent
identically...
Let X i ~ Unif(0, 1) for 1 <= i <= n be IID (independent
identically distributed) random variables. Let Y = max(X 1 , …, X n
). What is E(Y)?
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 X1, X2, . . . , Xn be iid random variables with pdf
f(x|θ) =...
Let X1, X2, . . . , Xn be iid random variables with pdf
f(x|θ) = θx^(θ−1) , 0 < x < 1, θ > 0.
Is there an unbiased estimator of some function γ(θ), whose
variance attains the Cramer-Rao lower bound?
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...
Let X1. ..., Xn, be a random sample from Exponential(β) with pdf
f(x) = 1/β(e^(-x/β)) I(0,...
Let X1. ..., Xn, be a random sample from Exponential(β) with pdf
f(x) = 1/β(e^(-x/β)) I(0, ∞)(x), B > 0 where β is an unknown
parameter. Find the UMVUE of β^2.