An affine image of an exponential density

Topic: Functions of random variables and limit theorems · more questions on this topic
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Let X\sim\operatorname{Exp}(1/2) and Y=1+2X. Find the density of Y and say where it is positive.

Worked answer (click to show)

The density of X is (1/2)e^{-x/2} for x>0. The map y=1+2x has inverse (y-1)/2 and derivative factor 1/2, so for y\ge 1

f_Y(y)=\frac{1}{2}\cdot\frac12\exp\bigl(-(y-1)/4\bigr)=\frac14\exp\bigl(-(y-1)/4\bigr).

The density is zero for y<1.