Probability and Statistics, Winter 2013
Grinshpan
Homework 3
1.
Consider a random sample X1 , . . . , Xn from the probability density function
f (x) = 0.5(1 + θx),
−1 ≤ x ≤ 1,
where θ is an unknown parameter. Show that θ̂ = 3X is an unbiased estimator of θ.
2.
Consider simple random sampling with replacement. Show that
n
1 X
s2 =
(Xi − X)2
n − 1 i=1
is an unbiased estimate of σ 2 , the population variance. Is s an unbiased estimate of σ?
3. Let 0.16, 0.56, 0.8, 0.41, 0.37 be a simple random sample from a large population data. Suggest
point estimates for the first, second, and third population moments. Estimate the standard error of the
population data.
4. A diagnostic test for a certain desease is applied to n disease-free individuals. Let X be the number
of false positives (the number of positive results among the n test results). Let p be the probability of a
false positive. Assume that only X is available rather than the actual sequence of test results. Form the
maximum likelihood estimator of p. If there are 3 false positive results among 20 test results, what is
the estimate? In this case, what is the maximum likelihood estimate that none of next five tests done on
disease-free individuals are positive?