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(a) Find k.
(b) Find P(X ? Y ). (c) Find the marginal pdf of X and Y.
(d) Find the joint cdf of X and Y.
(e) Are X and Y independent? Explicitly state your reason.


Section 2

 


 

STAT 330: Mathematical Statistics

 

Winter 2016

 

Assignment 2

 


 

Name:

 

ID:

 

Due on Feb 29th (Monday) 11:30am in drop boxes across the hall from MC 4065/4066.

 


 

1

 


 

1. [7 marks] Let

 


 

f (x, y) =

 


 

kxy 2 , if 0 ? x ? 2, 0 ? y ? 2

 

0,

 

otherwise

 


 

be the joint pdf of X and Y , where k is a positive constant.

 

(a) Find k.

 

(b) Find P (X ? Y ).

 

(c) Find the marginal pdf of X and Y .

 

(d) Find the joint cdf of X and Y .

 

(e) Are X and Y independent? Explicitly state your reason.

 


 

2. [7 marks] Let

 

f (x, y) =

 


 

3/2, if x2 ? y ? 1, 0 ? x ? 1

 

0,

 

otherwise

 


 

be the joint pdf of X and Y.

 

(a) Find P (0 ? X ? 1/2).

 

(b) Find P (X ? 1/2, Y ? 1/2).

 

(c) Are X and Y independent? Explicitly state your reason.

 

(d) Find the marginal pdf of X and Y .

 

(e) Find the conditional pdf of X given Y = y.

 

(f) Find E(X|Y = y).

 


 

3. [7 marks] Suppose (X1 , . . . , Xk ) ? M U LT (n, p1 , . . . , pk ). To answer the following questions,

 

you may use the facts: for i = j,

 

(1). the joint mgf of Xi and Xj is M(Xi ,Xj ) (t1 , t2 ) = (et1 pi + et2 pj + 1 ? pi ? pj )n .

 

(2). (Xi , Xj ) ? M U LT (n, pi , pj ).

 

(3). Xi ? BIN (n, pi ).

 

(4). Xi + Xj ? BIN (n, pi + pj ).

 

2

 


 

(a) Using the joint mgf, show that Cov(Xi , Xj ) = ?npi pj , for i = j.

 

pi

 

(b) Show that Xi |Xj = xj ? BIN (n ? xj , 1?pj ) for i = j.

 


 

(c) For i = j, let T = Xi + Xj . Show that Xi |T = t ? BIN (t, pi pi j ).

 

+p

 

4. [10 marks] Suppose X = (X1 , X2 )T ? BV N (µ, ?), where µ = (µ1 , µ2 )T and ? =

 


 

2

 

?1

 

??1 ?2

 

2

 

??1 ?2

 

?2

 


 

(a) Show that the joint mgf of X1 and X2 is

 

1

 

M (t1 , t2 ) = exp(µT t + tT ?t) for t ?

 

2

 


 

2

 


 

.

 


 

(Hint: First show that

 

(x?µ)T ??1 (x?µ)?2tT x = (x?µ??t)T ??1 (x?µ??t)?2µT t?tT ?t, where x = (x1 , x2 )T ).

 

2

 

2

 

(b) Use mgf to show X1 ? N (µ1 , ?1 ), and X2 ? N (µ2 , ?2 ).

 


 

(c) Find Cov(X1 , X2 ).

 

(d) Use mgf to show that

 

X1 and X2 are independent if and only if ? = 0.

 

(e) Let A be a 2 × 2 nonsingular matrix and b be a 2 × 1 vector. Use the mgf to show that

 

Y = AX + b ? BV N (Aµ + b, A?AT ).

 

5. [9 marks] Suppose X is a random variable following the standard normal distribution N (0, 1),

 

Y is a random variable, and the conditional pdf of Y given X = x is

 

(y?x)2

 

1

 

fY |X (y|x) = ? e? 2 , ?? < y < ?,

 

2?

 


 

where x is a real number.

 

(a) Find the joint pdf of X and Y .

 

(b) De?ne Z = Y ? X. Find the joint pdf of X and Z.

 

(c) Are X and Z are independent? Justify your answer.

 

(d) Find Cov(X, Y ).

 

(e) Find the marginal pdf of Y .

 


 

3

 


 

.

 


 

 


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