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MATH/STAT 3460, Intermediate Statistical Theory
Final Practice Questions
This Sample Final has more questions than the actual final, in order to cover
a wider range of questions.
1. Under a certain model, the number of is assumed to follow a censored
Poisson distribution with parameter λ. That is the probabilites of 0, 1
and 2 are e−λ , λe−λ and 1 − e−λ (1 + λ). The observed frequencies are :
(a) Use Newton’s method to find the Maximum Likelihood estimate for
λ. [Start with an initial estimate of 1, and perform 1 iteration.]
(b) Find the moment estimate for λ.
2. Every year a certain city is flooded with probability p. Based on data
from a record of whether or not the city flooded every year for the past
600 years. The maximum likelihood estimate for p is therefore p̂ = 600
where F is the number of years that the city is flooded.
An insurance company is interested in the probability that the city floods
at any time within the next 2 years. The probability of this is 1−(1−p)2 =
2p−p2 . The maximum likelihood estimate of this is therefore 300
What is the bias of this estimate?
3. Let X1 , ..., Xn be a random sample from a uniform distribution on [0, a].
The MLE for a is max(X1 , . . . , Xn ).
a) What is the bias of this estimator? [Hint: the expected value of a
random variable with cumulative distribution function F (x) is
b) Show that â = 2X̄ is an unbiased estimator for a.
c) Which of the above two estimators is better, based on the mean squared
error of these two estimators.
d) Is max(X1 , . . . , Xn ) also sufficient for a? Show your work for your
4. Let Y1 , ..., Yn be a random sample from the exponetial distribution fY (y; θ) =
θ , (y > 0). Compare the Cramér-Rao lower bound for f (y; θ) to the
variance of the maximum likelihood estimator for θ , θ̂ = Ȳ . Is Ȳ a best
estimator for θ?
5. You are conducting a survey. One of the questions is potentially sensitive
— the answer “YES” might be embarassing. To avoid embarrassment,
you attempt one of the following schemes:
• Ask the participant to roll a die (out of sight) and if it is 6, answer
“YES” regardless of the true answer. Otherwise answer the question
• Ask the participant to roll a die (out of sight) and if it is 6, give the
opposite of the true answer. Otherwise answer the question truthfully.
(a) What are the maximum likelihood estimates for the probability of
people who should really answer “YES” in each case?
(b) Are they both minimum variance unbiased estimators for the probability of people who should really answer “YES” ?
(c) Which scheme is more efficient?
6. Let Y1 , ..., Yn be a random
Pnsample of size n from a normal pdf having
µ = 0. Show that s2n = n1 i=1 Yi2 is a sufficient statistic and a consistent
estimator for σ 2 = V ar(Y ).
7. (a) A coin is tossed 10 times. The probability of coming up heads is p.
Show that the number of times heads comes up is a sufficient statistic for
(b) What is the probability of the sequence HHTTHHHTHTT given that
the number of heads is 5.
8. Let X denote the mean of a random sample of size n from a distribution
that has mean µ and variance σ 2 = 10. Find n so that the probability is
approximately 0.954 that the random interval (X − 21 , X + 12 ) includes µ.
9. Let X1 , X2 , ..., X9 be a random sample of size 9 from a distribution that
is N (µ, σ 2 ).
(a) If σ is known, find the length of a 95%√confidence interval for µ if this
interval is based on the random variable 9(X̄ − µ)/σ.
(b) If σ is unknown, find the expected value of the length of a 95% confidence
interval for µ if this interval is based on the random variable
9(X − µ)/S.
Hint: Write E(S) = (σ/ n − 1)E[((n − 1)S 2 /σ 2 )1/2 ].
(c) Compare these two answers.
10. Let the random variable X have the pdf f (x; θ) = (1/θ)e−x/θ , 0 < x <
∞, zero elsewhere. Consider the simple hypothesis H0 : θ = 2 and the
alternative hypothesis H1 : θ = 4. Let X1 , X2 denote a random sample
of size 2 from this distribution. Show that the best test of H0 against H1
may be carried out by use of the statistic X1 + X2 .
11. Let X1 , · · · , X10 be a random sample of size 10 from Binomial distribution
Bin(1, p), if we test the hypothesis H0 : p = 1/4 against H1 : p < 1/4 by
rejecting H0 if and only if 1 ≤ 1. Find the power function of this test
for 0 ≤ p ≤ 1/4.
12. Let us say the life of a tire in miles, say X, is normally distributed with
mean θ and standard deviation 5000. Past experience indicates that θ =
30, 000. The manufacturer claims that the tires made by a new process
have mean θ > 30, 000. It is possible that θ = 35, 000. Check his claim
by testing H0 : θ = 30, 000 against H1 : θ > 30, 000. We observe n
independent values of X, say x1 , ..., xn , and we reject H0 (thus accept
H1 ) if and only if x̄ ≥ c. Determine n and c so that the power function
π(θ) of the test has the values π(30, 000) = 0.01 and π(35, 000) = 0.98.
13. Let X1 , . . . , X50 be samples from a Normal distribution with mean µ and
variance σ 2 . Suppose that the sample mean is 2.3 and the sample variance
is 6.25. Using a likelihood ratio test, test the hypothesis that σ = 3 at the
5% significance level.
14. Let X1 , . . . , X20 be samples from a Poisson distribution with parameter
λ. Suppose that X1 + · · · + X20 = 123. Using a likelihood ratio test, test
the hypothesis that λ = 7 at the 5% significance level.
15. In a sample of 100 variables X1 , . . . , Xn , it is believed that the Xi are
independent samples from a binomial distribution with n = 2 and p = 0.1.
The following results are obtained:
Test the hypothesis using a chi-squared test at the 10% significance level.
16. A scientist is studying the effects of vitamin supplements on intelligence.
She takes 1000 subjects, and gives vitamin supplements to 500 of them
for a period of 3 months. Then she gives them all a standard test. The
results are below:
(a) Test the hypothesis that results were independent of whether the subjects had taken the vitamin supplement at the 5% significance level.
(b) Does this show that the vitamin supplement causes individuals to get
better scores in the tests. If not, give a possible alternative explanation.
17. Exponential model with conjugate prior distribution:
(a) Show that if y|θ is exponentially distributed with rate θ, then the
gamma prior distribution is conjugate for inferences about θ given an
independent and identically distributed sample of y values.
(b) The length of life of a light bulb manufactured by a certain process has
an exponential distribution with unknown rate θ. Suppose the prior distribution for θ is a gamma distribution with coefficient of variation 0.5.(The
coefficient of variation is defined as the standard deviation divided by the
mean.) A random sample of light bulbs is to be tested and the lifetime of
each obtained. If the coefficient of variation of the distribution of θ is to
be reduced to 0.1, how many light bulbs need to be tested?
18. Censored and uncensored data in the exponential model:
(a) Suppose y|θ is exponentially distributed with rate θ, and the marginal
(prior) distribution of θ is Gamma(α, β). Suppose we observe that y ≥
100, but do not observe the exact value of y. What is the posterior distribution, p(θ|y ≥ 100), as a function of α and β? Write down the posterior
mean and variance of θ.
(b) In the above problem, suppose that we are now told that y is exactly
100. Now what are the posterior mean and variance of θ?