A quadratic function of a uniform variable

Topic: Functions of random variables and limit theorems · more questions on this topic
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Let X be uniform on [1,3] and set Y=X^2-4. Find the density of Y, including the set on which it is zero.

Worked answer (click to show)

On [1,3] the map x\mapsto x^2-4 increases from -3 to 5. For y in that range,

\mathbb{P}(Y\le y)=\mathbb{P}\bigl(X\le\sqrt{y+4}\bigr)=F_X(\sqrt{y+4}).

The uniform density on [1,3] is 1/2, so

f_Y(y)=\frac{1}{2}\cdot\frac{1}{2\sqrt{y+4}}=\frac{1}{4\sqrt{y+4}}, \qquad -3<y<5,

and f_Y(y)=0 otherwise.