# Let $U \sim \textrm{Unif}(0, \pi/ 2)$. Find the PDF of $\sin(U)$.

This is almost the same as Suppose that X ∼ U ( $$− π/2$$ , $$π/2$$ ) . Find the pdf of Y = tan(X)., but making sure I am understanding the process:

Let $$U \sim \textrm{Unif}(0, \pi/ 2)$$. Find the PDF of $$\sin(U)$$.

\begin{align} F_Y(y) &= P(Y

That is the CDF. To find the PDF, take the derivative with respect to $$y$$ to get:

$$\frac{2}{\pi}\frac{1}{\sqrt{1-y^2}}$$

1. Is the work & logic correct?
2. Is the support of $$Y$$ the same as the support for U: $$[0,pi/2]$$?
• The support for $Y$ is $[0,1]$. Always good to confirm that the PDF integrates to 1 over $[0,1]$. In this case it does. You can compute the integral by substituting $y = \sin \theta$. – Aditya Dua Dec 8 '18 at 22:21