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Material Type: Exam; Class: Mathematical Statistics I; Subject: Statistics; University: University of Illinois - Urbana-Champaign; Term: Fall 2006;
Typology: Exams
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Friday, October 13, 2006
Closed book & Notes.
30 points 1. Suppose (X 1 , X 2 ) has space
X = {(x 1 , x 2 ) | x 1 > 0 , x 2 > 0 and x 1 + x 2 < 1 }
and pdf fX (x 1 , x 2 ) = 2. (a) Sketch the space X.
x
x
(b) Let Y 1 =
X 1 and Y 2 =
X 2. Find and sketch the space Y of (Y 1 , Y 2 ). Answer: Since X 1 goes from 0 to 1, Y 1 goes from 0 to 1, as does Y 2. But there is the extra constraint that X 1 + X 2 < 1, hence
Y = {(y 1 , y 2 ) | y 1 > 0 , y 2 > 0 , y 12 + y 22 < 1 }.
This space is a quarter-disk:
y
y
(c) Find the pdf of (Y 1 , Y 2 ).
Answer: The inverse function is x 1 = y^21 and x 2 = y 22 , so that the Jacobian is
Jg− 1 (y 1 , y 2 ) =
∣∣ ∣∣ ∣
2 y 1 0 0 2 y 2
∣∣ ∣∣ ∣ = 4y^1 y^2.
The pdf of (X 1 , X 2 ) is just 2, hence
fY (y 1 , y 2 ) = 8y 1 y 2.
(d) Are Y 1 and Y 2 independent? Why or why not?
Answer: No, the space in not a rectangle.
(e) What is the marginal space Y 1 of^ Y 1?
Answer: (0,1)
(f) What is the conditional space of Y 2 given Y 1 = y 1 for y 1 ∈ Y 1?
Answer: 0 < y^22 < 1 − y^21 , hence the conditional space is (0,
√ 1 − y 12 ).
(g) Find the marginal pdf of Y 1.
Answer: Integrate out y 2 :
fY 1 (y 1 ) =
∫ √ 1 −y (^21)
0
8 y 1 y 2 dy 2 = 8y 1
y 22 2
∣∣ ∣∣ ∣
1 −y^21
0
= 4y 1 (1 − y^21 ).
Answer: Convolutions yield:
fY (0) = fX 1 (0)fX 2 (0) =
fY (1) = fX 1 (0)fX 2 (1) + fX 1 (1)fX 2 (0) =
fY (2) = fX 1 (1)fX 2 (1) + fX 1 (2)fX 2 (0) =
fY (3) = fX 1 (2)fX 2 (1) =
25 points 3. Suppose X has the Gumbel(θ) distribution, which means that X has space R and distri- bution function FX (x) = e−θe
−x . Let Y = e−X^. (a) What is the space of Y? Answer: Y = (0, ∞).
(b) Find the pdf of Y. Answer: Using distribution functions,
FY (y) = P [Y ≤ y] = P [e−X^ ≤ y] = P [X ≥ − log(y)] = 1 − FX (− log(y)) = 1 − e−θy^ ,
hence fY (y) = F (^) Y′ (y) = θe−θy, which is the Exponential(θ) density. (You could also use Jacobians, being sure to first find the pdf of X.)
(c) Now suppose X 1 ,... , Xn are iid Gumbel(θ) random variables. Show that the distribu- tion of the maximum, X(n), is Gumbel, and give the parameter. [Hint: Find the distribution function of X(n). You can use the fact from the homework that the distribution function of the maximum is the distribution function of X to the nth^ power.] Answer: For any x, FX(n) (x) = (FX (x))n^ = e−nθe
−x , which is the distribution function of a Gumbel(nθ).
20 points 4. Recall that the Dirichlet(α 1 ,... , αK ) distribution is the distribution of
1 X 1 + · · · + XK
) ,
where the Xi’s are independent, Xi ∼ Gamma(αi, 1). Suppose that (Y 1 , Y 2 , Y 3 )^ ∼^ Dirichlet(1,^2 ,^1 ,^ 3). (a) What is the distribution of Y 1 Y 1 + Y 2
Answer: Write the Yi’s in terms of the Gamma Xi’s:
Y 1 Y 1 + Y 2
which is Beta(α 1 , α 2 ) = Beta(1, 2) ≡ Dirichlet(1, 2).
(b) What is the distribution of
Y 1 + Y 2 Y 1 + Y 2 + Y 3
Answer: Similarly, this is Beta(α 1 + α 2 , α 3 ) = Beta(3, 1).
(c) What is the distribution of ( Y 1 Y 1 + Y 2 + Y 3
) ?
Answer: And this is Dirichlet(1, 2 , 1).
