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Chapter 5: Q 5.5 (page 215)

Use the result that for a nonnegative random variableY,

EY=鈭�0鈭�PY>tdt

to show that for a nonnegative random variableX,

EXn=鈭�0鈭�nxn-1PX>xdx

Hint: Start with

EXn=鈭�0鈭�PXn>tdt

and make the change of variablest=xn.


Short Answer

Expert verified

Therefore, we鈥檝e shownEXn=鈭�0鈭�nxn-1PX>xdx=鈭�0鈭�nxn-1PX>tdtas required.

Step by step solution

01

Given information:

Use the result that for a non-negative random variableY,

EY=鈭�0鈭�PY>tdt

Hint: Start with

EXn=鈭�0鈭�PXn>tdt

and make the change of variablest=xn

02

Explanation:

By letting Y=Xnwe know that

EXn=鈭�0鈭�PXn>tdt

Making the change of variablest=xnanddt=nxn-1dx, we get that

EXn=鈭�0鈭�nxn-1PXn>xndx

03

Explanation:

Since u鈫�unis an increasing function of u鈮�0the events Xn>xnandX>xare identical. So we鈥檝e shown

EXn=鈭�0鈭�nxn-1PX>xdx=鈭�0鈭�nxn-1PX>tdtas required.

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Most popular questions from this chapter

(a) A fire station is to be located along a road of length A,A<鈭�. 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,

choose a so as to minimize localid="1646880570154" EX-awhen X is uniformly distributed over 0,A.

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

If xis uniformly distributed over (a,b), what random variable, having a linear relation with x, is uniformly distributed over (0,1)?

Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin 1000 times. If the coin lands on heads 525 or more times, then we shall conclude that it is a biased coin, whereas if it lands on heads fewer than 525 times, then we shall conclude that it is a fair coin.

Suppose that the life distribution of an item has the hazard rate function位(t)=t3,t>0. What is the probability that

(a)the item survives to age2?

(b)the item鈥檚 lifetime is between.4and1.4?

(c)a 1-year-old item will survive to age2?

IfXis uniformly distributed over (-1,1),find

(a)P{X>12}

(b)the density function of the random variableX.
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