MAST20005/MAST90058: Week 11 Problems
1. Let X(1) < ··· < X(5) be the order statistics of 5 independent observations from an exponential distribution that has a mean of θ = 3.
(a) Find the pdf of the sample median X(3). (b) Compute the probability that X(4) < 5.
(c) Determine Pr(1 < X(1)).
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2. Let X1, . . . , X10 be a random sample from a shifted exponential distribution with pdf
f(x | θ) = e−(x−θ), θ x < ∞.
(a) Show that Y = min(Xi) = X(1) is the maximum likelihood estimator of θ. (b) Find the pdf of Y .
(c) ShowthatE(Y)=θ+ 1 andthatY − 1 isanunbiasedestimatorofθ.
(d) Compute Pr(θ < Y < θ + c) and use it to construct a 95% confidence interval for θ. (e) Where have you seen this example before?
3. Let X(1) < ··· < X(n) be the order statistics of n independent observations from the uniform distribution Unif(0, 1).
(a) Find the pdf of X(1).
(b) Verify that E(X(1)) = 1 .
4. Let X have a Laplace distribution with pdf f(x | θ) = 12e−|x−θ|. (This is also known as a double exponential distribution, can you see why?) Suppose we have a random sample of n observations on X.
(a) Show that E(X) = θ and var(X) = 2. (Hint: ∞ z2e−zdz = 2) 0
(b) Consider the estimator, θˆ = X ̄ . Find its mean and variance. 1
(c) Consider the estimator, θˆ = Mˆ . Find its approximate mean and variance. 2
(d) Which estimator is better?
(e) What is the maximum likelihood estimator of θ?
5. The following times (in minutes) between tram arrivals were observed at a particular tram stop:
0.67, 2.46, 1.00, 8.89, 8.85, 28.45, 2.95,
2.36, 0.37, 5.66, 6.26, 1.80, 1.88, 4.66
Find an approximate 95% confidence interval for the median and state its exact confidence level. You may use the following information:
> pbinom(0:6, size = 14, prob = 0.5)
[1] 0.0001 0.0009 0.0065 0.0287 0.0898 0.2120 0.3953
> pbinom(13:7, size = 14, prob = 0.5)
[1] 0.9999 0.9991 0.9935 0.9713 0.9102 0.7880 0.6047
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