STAT 4101 Quiz 4, Spring 2021
First name: Last name:
Hawkid: @uiowa.edu
Please read the following instructions before you start the exam.
• There are three pages including the cover page.
• The quiz is available from Mar 26th, 2021 noon and due on Mar 29th, 2021 noon.
• There are 2 problems. The total is 10 ∗ 2 = 20 points.
• Show all work for each problem below. If you only present a correct answer
without stating how you get it, no credit will be given.
• Calculator ready solutions are not sufficient to obtain full credits. A numer- ical answer is needed if a problem asks for it.
• You must stop writing immediately after the invigilator signals the end of the exam.
• Any distribution of this material is prohibited.
1
1. (10 pts) Suppose that X1, . . . , Xn form a random sample from
=
∞ 2θ2 n
x3 dx θ2 n
2θ2 f(x;θ) = x3
θ
Solution: We show the convergence in probability directly based on the definition. For any ε > 0,
P (|Y1 − θ| > ε)
=P(Y1 >θ+εorY1 <θ−ε)
=P (Y1 > θ + ε) because Y1 is bounded below by θ
= (P (Xi > θ + ε))n
because of iid assumption
θ+ε
= (θ+ε)2 →0, asn→∞.
2
2. (10 pts) Let X1, . . . , Xn be a random sample from a Bernoulli distribution with parameter 0 < p < 1. (a) Use Chebyshev inequality to show that regardless the value of p,
̄ 1
P |X−p|≤√n ≥0.75.
Solution: Note that E(X ̄ ) = p and Var(X ̄ ) = p(1 − p)/n. By Chebyshev inequality, ̄ 1 E|X ̄−p|2 ̄
P |X−p|> √n ≤ (1/√n)2 =n·Var(X)=p(1−p)≤0.25.
It follows that P (|X ̄ − p| ≤ 1/√n) = 1 − P (|X ̄ − p| > 1/√n) ≥ 0.75. (b) Show X ̄2 is a consistent estimator of p2.
Solution: The weak law of large numbers implies that ̄P
X −→ p. By the continuous mapping theorem, we have
which says X ̄2 is a consistent estimator of p2.
End of the exam!
3
̄2P 2 X −→ p ,