Transform an exponential variable

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Problem

Let $X\sim\operatorname{Exponential}(2)$ under the rate parameterization and define $Y=2X$ . Derive the CDF and PDF of $Y$ , identify its distribution, and give its mean and variance.

Reveal answer or reference solution

For $y<0$ , $F_Y(y)=0$ . For $y\ge0$ ,

F_Y(y)=P(2X\le y)=P(X\le y/2)=1-e^{-2(y/2)}=1-e^{-y}

Differentiating gives $f_Y(y)=e^{-y}$ for $y\ge0$ and $0$ otherwise. Hence $Y\sim\operatorname{Exponential}(1)$ . Its mean is $1$ and its variance is $1$ .

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Transform an exponential variable

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