hw12 - STAT 418 HW12 Solutions Sections 6.2-6.5 15 20 21 23...

STAT 418
HW12 Solutions
Sections 6.2-6.5
15, 20, 21, 23, 27, 28, 32, 33; TE 9
15.
a. Let A denote the area of region R , then A =
R R
( x,y ) ∈ R
dydx. Because f ( x, y ) is a joint pdf,
1 =
Z Z
( x,y ) ∈ R
f ( x, y ) dydx =
Z Z
( x,y ) ∈ R
cdydx = c
Z Z
( x,y ) ∈ R
dydx = cA.
Therefore c = 1 /A.
b.
f ( x, y ) =
1
4
, x ∈ ( - 1 , 1) , y ∈ ( - 1 , 1)
1

21.
a. f ( x, y ) ≥ 0 for all x, y .