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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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