Stat 776 Homework Solution 5: Probability Distributions and Linear Regression - Prof. Xiao, Assignments of Data Analysis & Statistical Methods

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Stat 776 HW

1. X =

( X

1 X 2

)

∼ N

(( 1. 5

0

)

,

( 1

))

. Find P (X 1 < 0).

X 1 ∼ N (1. 5 , 12 ) P (X 1 < 0) = P (Z < − 1 .5) = 0. 0668

2. p201 4.4 [5th edition: p203 4.4] (a) Find the distribution of 3X 1 − 2 X 2 + X 3 3 X 1 − 2 X 2 + X 3 = (3, − 2 , 1)X ∼ N



(3, − 2 , 1)





− 3



 , (3, − 2 , 1)













− 2









= N (13, 32 )

(b) Find a =

( (^) a 1 a 2

)

such that X 2 and X 2 −a′

( X

1 X 3

)

are independent

(0, 1 , 0)













−a 1 1 −a 2



 = 0

−a 1 + 3 − 2 a 2 = 0 a =

( (^3) − 2 a 2 a 2

)

The solution is not unique, a =

( 1

)

for example is one solution.

3. X ∈ Rp×n^ is a random matrix, and X ∼ N (M, Σ). A ∈ Rn×n^ is a symmetric idempotent matrix. Find and list all independent pairs from XA, XAX′, X(I − A) and X(I − A)X′.

(a) XA and X(I − A) are independent A′(I − A) = 0. So XA and X(I − A) are independent (b) XA and X(I − A)X′^ are independent (I − A)′^ = I − A and A′(I − A) = 0. So XA and X(I − A)X′^ are independent (c) XAX′^ and X(I − A) independent A′^ = A and (I − A)′A = 0. So XAX′^ and X(I − A) are independent (d) XAX′^ and X(I − A)X′^ are independent A′^ = A, (I −A)′^ = I −A and A(I −A) = 0. So XAX′^ and X(I −A)X′ are independent