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Chapter 5: Q.5.3 (page 212)

Consider the function

f(x)=C(2x−x3)0<x<520otherwise

Could fbe a probability density function? If so, determine

C. Repeat if f(x)were given by

localid="1646632841793" f(x)=C(2x−x3)0<x<520otherwise

Short Answer

Expert verified

Whenc=0,±2thenf(x)<0

Step by step solution

01

Given Information

f(x)=C(2x−x3)0<x<520otherwise

02

Explanation

2x-x3=-x(x-2)(x+2)

c=0

∫-∞∞f(x)dx=0,∫-∞∞f(x)dx=1

c>0,f(x)<0,x∈(2,52)

03

Explanation

c=2

f(x)=2(2x-x3)=4x-2x3

f(x)<0,x∈(2,52)

f(1.5)=4×15-2×(1.5)3=-0.75

c<0,f(x)<0(0,2)

04

Explanation

c=-2

f(x)=-2(2x-x3)

role="math" localid="1646639321144" =-4x+2x3

f(1)=-4+(2×1)=-2

f(x)≥0

05

Explanation

f(x)=C(2x−x3)0<x<520otherwise

2x-x2=-x(x-2)

c=0,∫-∞∞f(x)dx=0

c>0,f(x)<0,x∈(2,52)

06

Explanation

x=2.2

f(2.2)=4(2.2)-2(2.2)=-0.88

c<0,x∈(0,2)

c=-2,f(x)=-4x+2x2

f(1)=-4+2=-2<0

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Most popular questions from this chapter

The lung cancer hazard rateλ(t)of a t-year-old male smoker is such that

λ(t)=.027+.00025(t−40)2t≥40

Assuming that a 40-year-old male smoker survives all other hazards, what is the probability that he survives to

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(a) A fire station is to be located along a road of length A,A<∞. If fires occur at points uniformly chosen on localid="1646880402145" 0,A, where should the station be located so as to minimize the expected distance from the fire? That is,

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(b) Now suppose that the road is of infinite length— stretching from point 0outward to ∞. If the distance of a fire from point 0is exponentially distributed with rate λ, where should the fire station now be located? That is, we want to minimize EX-a, where X is now exponential with rate λ.

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