## Transcribed Text

Problem 1 Let x be one observation from a N(0,0²) population. Is |X| a sufficient statistic?
Problem 2 Let X1. X= be independent random variables with densities
= 0
Prove that T = is a sufficient statistic for 0.
Problem 3 One observation is taken on a discrete random variable x with pmf f(x|0), where
OE {1,2,3}. Find the MLE of 0.
x f(x|1) f(x|2) f(x)3)
1/3
1/4
1
1/3
1/4
2
1/4
1/4
3
1/6
1/4
1/2
4
1/6
1/4
Problem 4 Let X1
X. be i.i.d with Xi n Poisson(A), r > 0. Compute the ML estimator
for A.
Problem 5 Assume that a gas station needs at least a > 0 minutes for an oil change, where a
is some constant. The actual time that an oil change requires varies from case to case. Suppose
you want to model the required time for an oil change by a random variable x = Z+a, where
Z is a random variable that satisfies Z 0. Suppose that Z N Exponential (1), i.e., the density
of Z is given by
(a) What is the density of X?
1
(b) Suppose that you have 10 independent observations of times required for an oil change:
1.2,3.1,3.6,4.5,5.1,7.6,4.4,3.5,3.8,4
Based on these data, compute the ML estimate for a.
Problem 6 During the preparation for an exam Bob and Anna communicate via e-mail to
clarify arising questions. Due to the time points that Anna receives e-mails from Bob she is
afraid that he works more than she does. The time points (on a 24 hour scale) of the last ten
e-mails are 10.55, 14.9, 11.2, 18.85, 9.75, 11.5, 16.1, 14.4,9.2, 12.95;
Based on these data Anna wants to estimate the daily working time of Bob. For this pur-
pose she assumes that Bob starts studying each day at a fixed time a and works until a fixed
time b > a. Furthermore, she assumes that the time points when Bob sends e-mails are inde-
pendent and uniformly distributed on the interval [a, bj. Can you help Anna and estimate for
her the parameters a and b with
(a) ML estimation;
(b) The method of moments, i.e., find those parameters a, b for which the mean and the vari-
ance of a Ua, b] distribution coincide with the sample average and the sample variance,
respectively.
What do you think about these estimation procedures?
2

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