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Chapter 5: Q.5.15 (page 215)

If X is an exponential random variable with parameter 位, and c > 0, show that cX is exponential with parameter 位/c

Short Answer

Expert verified

With differentiating, we have that,

ddxFcX(x)=ddxFX(xc)鈰�1c=1cfX(xc)=1c位e鈭�位xc

Finally, we have obtained that forx鈮�0we have that

fcX(x)=位ce鈭�位cx

and otherwise it is equal to zero. Hence, we have proved thatcX~Expo鈦�(位c)

Step by step solution

01

Given Information

If X is an exponential random variable with parameter 位, and c > 0, show that cX is exponential with parameter 位/c

02

Explanation

Since we know thatP(X鈮�0)=1, we have that

P(cX鈮�0)=P(X鈮�0)=1

socXis also non- negative random variable. The distribution ofcXis

FcX(x)=P(cX鈮�x)=P(X鈮�xc)=FX(xc)

So, with differentiating, we have that,

ddxFcX(x)=ddxFX(xc)鈰�1c=1cfX(xc)=1c位e鈭�位xc

Finally, we have obtained that forx鈮�0we have that

fcX(x)=位ce鈭�位cx

and otherwise it is equal to zero. Hence, we have proved thatcX~Expo鈦�(位c)

03

Final Answer

With differentiating, we have that,

ddxFcX(x)=ddxFX(xc)鈰�1c=1cfX(xc)=1c位e鈭�位xc

Finally, we have obtained that forx鈮�0we have that

fcX(x)=位ce鈭�位cx

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Most popular questions from this chapter

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with 渭=4and蟽2=4, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters(6,9). A point is randomly chosen on the image and has a reading of5. If the fraction of the image that is black is伪, for what value of伪 would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region?

Let Y=X-饾湀饾浖.

Show that if X is a Weibull random variable with parameters 谓, 伪, and 尾, then Y is an exponential random variable with parameter 位 = 1 and vice versa.

Let X be a continuous random variable having cumulative distribution function F. Define the random variable Y by Y = F(X). Show that Y is uniformly distributed over (0, 1).

Let Z be a standard normal random variable Z, and let g be a differentiable function with derivative g'.

(a) Show that E[g'(Z)]=E[Zg(Z)];

(b) Show that E[Zn+1]=nE[Zn-1].

(c) Find E[Z4].

Suppose that the height, in inches, of a 25-year-old man is a normal random variable with parameters role="math" localid="1646741074533" 渭=71and蟽2=6.25. What percentage of 25-year-old men are taller than 6 feet, 2 inches? What percentage of men in the 6-footer club are taller than 6 feet, 5 inches?
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