AMS 205 – Homework 6
due in class, Tuesday November 16
1. Let X1,...,Xnbe a random sample from each of the distributions having the following
probability density functions:
(a) f(x;θ) = θx(θ−1),0<x<1,0<θ<∞, zero elsewhere
(b) f(x;θ) = 1
2exp(−|x−θ|), where the ranges of xand θare the real line
In each case, find an estimator of θby the method of moments and show that it is consistent
(hint: to show consistency, consider using Slutsky’s theorem and the fact that if a sequence
converges in distribution to a constant, then it converges in probability to that constant).
2. Let Y1< Y2< . . . < Ynbe the order statistics of a random sample of size nfrom the uniform
distribution of the continuous type over the closed interval [θ−ρ, θ +ρ]. Find the maximum
likelihood estimators for θand ρ. Are these two unbiased estimators?
3. Let the observed value of the mean Xof a random sample of size 18 from a distribution that
is N(µ, 80) be 81.6. Find a 95% confidence interval for µ.
4. Let a random sample of size 17 from the normal distribution N(µ, σ 2) yield ¯x= 4.7 and
s2= 5.76. Determine a 90 percent confidence interval for µ.
5. Suppose it is known that a random variable Xhas a Poisson distribution with parameter λ.
A sample of 200 observations from this population has a mean equal to 3.4. Construct an
approximate 90% confidence interval for λ.
6. Let two independent random samples, each of size 10, from two normal distributions N(µ1, σ2)
and N(µ2, σ2) yield ¯x= 4.8, (s1)2= 8.64, ¯y= 5.6, (s2)2= 7.88. Find a 95% confidence
interval for µ1−µ2.