Description
1. Show that the following p.d.f’s belong to the exponential family
(a) Poisson;
(b) Binomial;
(c) Negative Binomial;
(d) Gamma(θ; k), where k > 0 is konwn;
(e) N(θ; 1);
(f) N(0; θ).
2. Let X1, …, Xn denote a random sample of size n > 2 from a distribution with p.d.f. f(x; θ) =
θe−θx, 0 < x < ∞, zero elsewhere, and θ > 0. Then Y =
Pn
i=1 Xi
is a sufficient statistic for
θ. Prove that (n − 1)/Y is the best statistic for θ.
3. Let X1, X2, …, Xn be a random sample from f(x|θ) = θe−θx, x > 0, where θ > 0.
(i) Find a 100(1 − α)% confidence interval for the mean 1/θ;
(ii) Find a 100(1 − α)% confidence interval for the variance 1/θ2
;
(iii) What is the probability that these intervals cover the true mean and true variance,
simultaneously?
(iv) Find a 100(1 − α)% confidence interval for τ = e
−θ
;
4. Suppose X1, …, Xn are independent random variable with distribution B(1, p).
(a) Find the maximum likelihood estimator of θ = (1 − p)
2
;
(b) Show that ˆθ is an unbiased estimator of θ, where ˆθ = 1 if X1 + X2 = 0 and ˆθ = 0
otherwise;
(c) Find the best estimator for θ.
5. In comparing the times until failure (in hours) of two different types of light bulbs, we obtain
the sample characteristics n1 = 45, x¯ = 984, s
2
x = 8742 and n2 = 52, y¯ = 1121, s2
y = 9411.
Find an approximate 90% confidence interval for the difference of the two population means.
Interpret the result and explain why we can use the normal table here despite the fact that
the distribution of individual failure times is probably exponential or Weibull.
6. The effectiveness of two methods of teaching statistics is compared. A class of 24 students is
randomly divided into two groups and each group is taught according to a different method.
Their test scores at the end of the semester show the following characteristics:
n1 = 13, x¯ = 74.5, s2
x = 82.6, n2 = 11, y¯ = 71.8, s2
y = 112.6
Assuming underlying (approximate) normal distributions with σ1 = σ2, find a 95% confidence interval for µ1 − µ2. Compute 90% confidence interval for σ1/σ2.
