Let X1, X2, . . . be iid random variables following a uniform
distribution on the interval [0, θ]. Show that max(X1, . . . , Xn)
→ θ in probability as n → ∞
Suppose X1, X2, ..., Xn is a random sample from a Poisson
distribution with unknown parameter µ.
a. What is the mean and variance of this distribution?
b. Is X1 + 2X6 − X8 an estimator of µ? Is it a good estimator?
Why or why not?
c. Find the moment estimator and MLE of µ.
d. Show the estimators in (c) are unbiased.
e. Find the MSE of the estimators in (c).
Given the frequency table below:
X 0...
Let X1, X2, . . . , Xn be iid following a uniform distribution
over the interval (θ, 2θ) (θ > 0).
(a) Find a method of moments estimator of θ.
(b) Find the MLE of θ.
(c) Find a constant k such that E(k ˆθ) = θ.
(d) By using the Rao-Blackwell, which estimators of (a) and (b)
can be improved?
2. Let X1, X2, . . . , Xn be independent, uniformly distributed
random variables on the interval [0, θ]. (a) Find the pdf of X(j) ,
the j th order statistic. (b) Use the result from (a) to find
E(X(j)). (c) Use the result from (b) to find E(X(j)−X(j−1)), the
mean difference between two successive order statistics. (d)
Suppose that n = 10, and X1, . . . , X10 represents the waiting
times that the n = 10...
Let X1, X2, X3, . . . be independently random variables such
that Xn ∼ Bin(n, 0.5) for n ≥ 1. Let N ∼ Geo(0.5) and assume it is
independent of X1, X2, . . .. Further define T = XN
.
(a) Find E(T) and argue that T is short proper.
(b) Find the pgf of T.
(c) Use the pgf of T in (b) to find P(T = n) for n ≥ 0.
(d) Use the pgf of...
Let X1,…, Xn be a sample of iid N(0, ?) random
variables with Θ=(0, ∞). Determine
a) the MLE ? of ?.
b) E(? ̂).
c) the asymptotic variance of the MLE of ?.
d) the MLE of SD(Xi ) = √ ?.
Let X1,…, Xn be a sample of iid random variables with pdf f (x;
?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ.
Determine
a) L(?1, ?2).
b) the MLE of ?⃗ = (?1, ?2).
c) E(? ̂ 2).
Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ).
Consider the hypotheses H0 : µ = µ0 and H1 : µ (not equal)= µ0
and the test statistic (X bar − µ0)/ (S/√ n). Note that S has been
used as σ is unknown.
a. What is the distribution of the test statistic when H0 is
true?
b. What is the type I error of an α−level test of this type?
Prove it.
c. What is...
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, find a two dimensional sufficient statistic for (a, b)