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STAT 441: Homework 3 Due: Monday, 04/10/2023 by 11:59 pm
1. Let {!, ", … , #} be a random sample from the exponential distribution with parameter .
Show that = ∑$ is a sufficient statistic for .
2. Let !, ", … , # be a i.i.d. random sample from a continuous uniform distribution over (, )
where both and are unknown.
(a) Find the Method of Moments (MOM) estimators for and .
(b) Suppose a random sample of 4 numbers are observed:
{1.4, 9.1, 5.8, 8.4}
Estimate the values of and using these sample data.
3. Let {!, ", … , #} be a random sample of size from a chi-square distribution "(), where
the number of degrees of freedom, , is unknown.
(a) Find the method of moment (MOM) estimator of .
(b) Suppose we observed a random sample of 10 values !, ", … , !% from that chi-square
distribution such that 4 $!%$&! = 21.5.
Estimate the value of based on this sample and the estimator you found in part (a).
4. Suppose a discrete random variable has a p.m.f. (|), where ∈ 1,2,3. The following
table provides the family of p.m.f.
(| = 1) (| = 2) (| = 3)
0 1/6 1/4 1/5
1 1/4 1/4 1/5
2 0 1/4 1/5
3 1/2 1/8 1/5
4 1/12 1/8 1/5
(a) Suppose = 2 is observed. What is the Maximum Likelihood Estimator (MLE) of ?
(b) Suppose a random sample of size 3, which is (!, ", ') = (1,3,4) is observed. What is
the MLE of ?