STAT 4101 Quiz 2, 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 from the noon on Feb 26th, 2021 to the noon on Mar 1st, 2021.
• The deadline
• 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 X1,X2,…,X10 is a random sample from uniform distribution on (0,θ), and the corre- sponding order statistics are Y1, Y2, . . . , Y10, from the smallest to the largest.
• What is the expectation of Y5?
• If we use 2Y5 to estimate θ, then what is the bias?
Solution: Let X ̃i = Xi/θ, then we know X ̃1, X ̃2, . . . , X ̃10 forms a random sample from uniform distribution (0, 1). We further observe Y ̃i = Yi/θ since the division by θ does not change the order.
Since Y ̃j’s are order statistics of independent uniformly distributed random variables, we have shown on Slide 8 of Chapter 4 Part 2 that each Y ̃j follows Beta distribution. To show this, we have
In this example, we have j = 5, so
and thus
The bias of 2Y5 is thus
n! yj−1(1 − y)n−j (j − 1)!(n − j)!
E(Y ̃5) = 5 10+1
E(Y5) = E(Y ̃5)θ = 5 θ. 10+1
E(2Y5) − θ = 10θ − θ = − 1 θ. 11 11
gY ̃ (y) = j
yj−1(1 − y)n−j
so each Y ̃j follows a Beta distribution (j, n − j + 1) with the mean j/(n + 1).
=
Γ(n + 1) Γ(j)Γ(n − j + 1)
2
2. (10 pts) Let X and Y denote independent random variables with respective probability density functions fX(x)=2x,0
whose inverse functions are X = V and Y = U. Thus
where
where
Therefore
f(2) (u, v) = fX,Y (v, x)∥J2∥ = 6u2v, U,V
du du
dx dy |J2| =
End of the exam!
dx dy 1 0
|J1| = = = 1.
dv dv 0 1 dx dy
U = Y, and V = X,
dv dv
dx dy
0 1
= = −1. 1 0
fU,V(u,v)=f(1) (u,v)+f(2) (u,v)=6uv(u+v), 0