Problem 1.
Suppose that x1; x2; x3; x4; x5, is a sample drawn from the exponential distribution
f(x; ) :=
(
e x if x 0
0 otherwise;
where is an unknown parameter.
(a) [2 points] Find an expression for the joint likelihood function f(x1; x2; x3; x4; x5; ).
(b) [3 points] Is x a sucient statistic for ? Explain your answer.
(c) [3 points] Now suppose that x1 = 5; x2 = 3; x3 = 4; x4 = 2; x5 = 2. Find the maximum
likelihood estimate of .
Problem 2.
Let X1; : : : ;X120 be a collection of i.i.d. random variables and let Y1; : : : ; Y120 be the associated
order statistics.
(a) [4 points] Find a value of j such that the interval [Y30..j ; Y30+j ] is an approximate 95%
condence interval for the 1
4 quartile of the distribution.
Problem 3.
A political candidate hires a polling company to determine their level of support, i.e. the
probability p that a random person will vote for them. The polling company will contact n
people and collect the data x1; : : : ; xn where xi = 1 if person i supports the candidate and
xi = 0 if the person does not support the candidate. Thus, the people contacted by the polling
company can be treated as i.i.d. b(1; p) random variables. Let x be the sample mean of the
data.
(a) [3 points] How many people does the polling company need to contact in order to guar-
antee that (x :02; x + :02) is a 95% condence interval for p? (Use the overestimate
p(1 p) 1
4 to help estimate the variance).
(b) [3 points] The client believes that their true level of support is ^p = :7. How many fewer
people need to be contacted (compared to part (a)) if you instead estimate the variance
by replacing p(1 p) with ^p(1 ^p) = :21 instead?
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