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 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?
Problem 4.
Suppose you have a normal distribution N(; 30) where is unknown. You want to test the
null hypothesis = 4 against the alternative hypothesis < 4. Suppose you create a
where you collect n data points x1; : : : ; xn and reject the null hypothesis if the sample mean
x < c for some cuto value c.
(a) [3 points] Let K() be the power function of the at some value 2 R. Give an
expression in terms of n; c; for K() (note your nal expression should either be in terms
of the error function erf OR it should be the probability of an N(0; 1) random variable
belonging to a certain region).
(b) [4 points] Find values for c and n that guarantee that the has at most :005 type 1
error and at most :01 type 2 error when = 2
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