Chapter 5: Q.5.15 (page 215)

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

Short Answer

Expert verified

With differentiating, we have that,

$\frac{d}{d x} F_{c X} (x) = \frac{d}{d x} F_{X} (\frac{x}{c}) \cdot \frac{1}{c} = \frac{1}{c} f_{X} (\frac{x}{c}) = \frac{1}{c} λ e^{- λ \frac{x}{c}}$

$Finally, we have obtained that for x \geq 0 we have that$

f_{c} X (x) = \frac{λ}{c} e^{- \frac{λ}{c} x}

$and otherwise it is equal to zero. Hence, we have proved that c X ~ Expo (\frac{λ}{c})$

Step by step solution

Given Information

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

Explanation

$Since we know that P (X \geq 0) = 1, we have that$

$P (c X \geq 0) = P (X \geq 0) = 1$

$so c X is also non- negative random variable. The distribution of c X is$

$F_{c X} (x) = P (c X \leq x) = P (X \leq \frac{x}{c}) = F_{X} (\frac{x}{c})$

So, with differentiating, we have that,

$\frac{d}{d x} F_{c X} (x) = \frac{d}{d x} F_{X} (\frac{x}{c}) \cdot \frac{1}{c} = \frac{1}{c} f_{X} (\frac{x}{c}) = \frac{1}{c} λ e^{- λ \frac{x}{c}}$

$Finally, we have obtained that for x \geq 0 we have that$

$f_{c} X (x) = \frac{λ}{c} e^{- \frac{λ}{c} x}$

$and otherwise it is equal to zero. Hence, we have proved that c X ~ Expo (\frac{λ}{c})$

Final Answer

With differentiating, we have that,

$\frac{d}{d x} F_{c X} (x) = \frac{d}{d x} F_{X} (\frac{x}{c}) \cdot \frac{1}{c} = \frac{1}{c} f_{X} (\frac{x}{c}) = \frac{1}{c} λ e^{- λ \frac{x}{c}}$

$Finally, we have obtained that for x \geq 0 we have that$

$f_{c} X (x) = \frac{λ}{c} e^{- \frac{λ}{c} x}$

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

Short Answer

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Explanation

Final Answer

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