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STAT 410 HW #2 Answers
Due Wednesday, September 24, 4PM You can turn it in to my office, 116B IH, or mailbox in 101 IH.
- Suppose Y 1 ,... , Yn are uncorrelated random variables with the same mean μ and same variance σ^2. Let Y = (Y 1 ,... , Yn)′.
(a) Write down E[Y ] and Cov[Y ].
Answer:
E[Y ] =
E[Y 1 ]
E[Y 2 ]
E[Yn]
μ μ .. . μ
= μ (^1) n,
where 1n is the n × 1 vector of all 1’s.
Cov[Y ] =
V ar[Y 1 ] Cov[Y 1 , Y 2 ] · · · Cov[Y 1 , Yn] Cov[Y 2 , Y 1 ] V ar[Y 2 ] · · · Cov[Y 2 , Yn] .. .
Cov[Yn, Y 1 ] Cov[Yn, Y 2 ] · · · V ar[Yn]
σ^2 0 · · · 0 0 σ^2 · · · 0 .. .
0 0 · · · σ^2
= σ^2 In,
where In is the n × n identity matrix.
(b) For an n × 1 vector a, show that a′Y has mean μ
∑ ai and variance σ^2 ‖a‖^2.
Answer:
E[a′Y ] = a′E[Y ] = a′μ (^1) n = μa′ (^1) n = μ
∑^ n
i=
ai.
V ar[a′Y ] = a′Cov[Y ]a = σ^2 a′Ina = σ^2 a′a = σ^2 ‖a‖^2.
(c) If a =^1 n 1 n (where 1n is the^ n^ ×^ 1 vector of all 1’s), then what is^ a′Y^ normally called? Use part (b) to find its mean and variance.
Answer:
a′Y = (
n
(^1) n)′Y =
n
∑^ n
i=
Yi = Y ,
the sample mean. From (b),
E[Y ] = μ
n
∑^ n
i=
1 = μ
n n
= μ,
and
V ar[Y ] = σ^2 ‖
n
(^1) n‖^2 = σ^2
∑^ n
i=
( 1 n
) 2 = σ^2 n
n^2
σ^2 n
2.1. (There were two problems 2.) Now suppose Y 1 , Y 2 , Y 3 and Z are uncorrelated, where the Yi’s all have mean μ and variance σ^2 > 0, and Z has mean 0 and variance τ 2 > 0. Let X 1 = Y 1 + Z, X 2 = Y 2 + Z, and X 3 = Y 3 + Z. Find A so that
X ≡
X 1
X 2
X 3
=^ A
Y 1
Y 2
Y 3
Z
.
Answer:
A =
(a) Find the mean and covariance matrix of X. What are the correlations of the Xi’s? Are the Xi’s independent?
Answer:
E[X] = A E
Y 1
Y 2
Y 3
Z
=
μ μ μ 0
=
μ μ μ
= μ 1
Cov[X] = A Cov
Y 1
Y 2
Y 3
Z
A′
σ^2 0 0 0 σ^2 0 0 0 σ^2 0 0 0 τ 2
σ^2 + τ 2 τ 2 τ 2 τ 2 σ^2 + τ 2 τ 2 τ 2 τ 2 σ^2 + τ 2
The correlations of the Xi’s are all τ 2 /(τ 2 + σ^2 ). These are positive, so the Xi’s are not independent.
x
y
(c) Find the (marginal) pdf of X.
Answer:
fX (x) =
∫
Yx
f (x, y)dy =
∫ √ 1 −x 2
− √ 1 −x^2
π
dy =
π
1 − x^2.
(d) Find the pdf of U = X^2. It should be a Beta(α, β). What are α and β?
Answer:
FU (u) = P [U ≤ u] = P [X^2 ≤ u] = P [−
u ≤ X ≤
u] =
∫ √u
−√u
fX (x)dx.
So, for u ∈ U = (0, 1),
fU (u) = F (^) U′ (u) = fX (
u)
u
− fX (−
u)
u
u
1 − u.
This is the pdf of a Beta(^12 , 32 ).
- Suppose X 1 , X 2 and X 3 are independent, all with distribution P [Xi = −1] = P [Xi = +1] = 12. Let Y 1 = X 1 X 2 , Y 2 = X 1 X 3 and Y 3 = X 2 X 3.
(a) What are the space Y and pmf fY of Y = (Y 1 , Y 2 , Y 3 )′?
Answer: The space of X is X = {− 1 , +1} × {− 1 , +1} × {− 1 , +1}, so
(x 1 , x 2 , x 3 ) (y 1 , y 2 , y 3 ) (+1, +1, +1) (+1, +1, +1) (+1, +1, −1) (+1, − 1 , −1) (+1, − 1 , +1) (− 1 , +1, −1) (+1, − 1 , −1) (− 1 , − 1 , +1) (− 1 , +1, +1) (− 1 , − 1 , +1) (− 1 , +1, −1) (− 1 , +1, −1) (− 1 , − 1 , +1) (+1, − 1 , −1) (− 1 , − 1 , −1) (+1, +1, +1)
Thus the space of is Y = {(1, 1 , 1), (1, − 1 , −1), (− 1 , 1 , −1), (− 1 , − 1 , 1)}. The pmf is
fY (y 1 , y 2 , y 3 ) =
for (y 1 , y 2 , y 3 ) ∈ Y, because each element in Y corresponds to two in X , e.g.,
fY (1, 1 , 1) = P [X = (1, 1 , 1)] + P [X = (− 1 , − 1 , −1)] =
(b) Are the Yi’s jointly independent?
Answer: No. The marginal spaces of the Yi’s are {− 1 , +1}, and Y = {− 1 , +1} × {− 1 , +1} × {− 1 , +1}. There are only 4 elements in Y, not 8. For example, (− 1 , 1 , 1) ∈ {− 1 , +1} × {− 1 , +1} × {− 1 , +1} but not in Y. Y always has an even number of −1’s.
(c) Find the marginal space and pmf of (Y 1 , Y 2 ). Are Y 1 and Y 2 independent? What about Y 1 and Y 3? Or Y 2 and Y 3?
Answer: Just take the elements of Y and erase the last component, to get the space {(1, 1), (1, −1), (− 1 , 1), (− 1 , −1)}. The pmf is fY 1 ,Y 2 (y 1 , y 2 ) = 14 , since there is a one-to-one correspondence with Y. Yes, Y 1 and Y 2 are independent, because the space is a rectangle {− 1 , +1} × {− 1 , +1}, and the pmf factors: fY 1 ,Y 2 (y 1 , y 2 ) = 12 × 12. Similarly, Y 1 and Y 3 are independent, and Y 2 and Y 3 are independent.
(d) What is the distribution of Y 1 Y 2 Y 3?
Answer: Y 1 Y 2 Y 3 is always 1, because an even number of them is −1. (Or notice that Y 1 Y 2 Y 3 = X 12 X 22 X 32 = 1.) So the space is { 1 } and the pmf is f (1) = 1.
4 (a). Find the moment generating function of Y ∼ P oisson(λ). For what values of t is it finite?
