The square 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 [0,1] and Y=X^2. Find the density of Y.

Worked answer (click to show)

For 0<y\le 1, \mathbb{P}(Y\le y)=\mathbb{P}(X\le\sqrt{y})=\sqrt{y}. Differentiating gives f_Y(y)=1/(2\sqrt{y}) on (0,1], and 0 elsewhere.