Question

The weekly demand for propane gas (in 1000s of gallons) from a particular facility is an rv X with the following pdf.

f(x) =

2 1 − 1 x2		1 ≤ x ≤ 2
0		otherwise

(a) Compute the cdf of X.

F(x) =	0	x < 1
		1 ≤ x ≤ 2
	1	2 > x

(b) Obtain an expression for the (100p)th percentile.
η(p) =

What is the value of ? (Round your answer to three decimal places.)


(c) Compute E(X) and V(X). (Round your answers to four decimal places.)

E(X)	=  thousand gallons
V(X)	=  thousand gallons squared

(d) If 1.6 thousand gallons are in stock at the beginning of the week and no new supply is due in during the week, how much of the 1.6 thousand gallons is expected to be left at the end of the week? [Hint: Let h(x) = amount left when demand = x.] (Round your answer to three decimal places.)
thousand gallons

Answer 1

a)

a)cdf F(x)=∫12 f(x) dx = 2*(x+1/x)-4    for 1<x<2

b)

F(x)=2(x+1/x)-4 ; therefore 2*(xp+1/xp)-4 =p ; solving it:ηp =(1/4)*(4+p+√(p2+8p))   ;

for median at p=0.5 , x=1.640

c)

E(x)=∫12 xf(x) dx = 2(x2/2-ln(x))|21 =2*(3/2-ln(2))=3-ln(4)=				1.6137 thousand gallons
E(x2)=∫12 x2f(x) dx = 2(x3/3-x)|21 =8/3=			2.6667
Var(X)=E(X2)-(E(X))2=			0.0626 thousand gallons

d)

E(h(x))=E(1.6-x)=∫11.6 (1.6-x)*f(x) dx = 1.6*∫f(x)dx-2(x2/2-ln(x))|1.61 =

0.1000