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The following problem is from the Schaum book called "Theory and Problems of Probability, Random Variables, and Random Processes":

Let $Y = \sin X$, where $X$ is uniformly distributed over $(0, 2)$. Find the pdf of $Y$.

Here is my solution: $$ P(Y \le y_0) = P(\sin X \le y_0 ) = P( X \le \arcsin y_0 ) = \int_0^{\arcsin y_0} \frac 1{ 2\pi } dx $$ When I evaluate that integral I get a function with differs from the book's answer by a factor of 2. I am hoping that somebody here can tell me what I am missing.

Bob

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  • $\begingroup$ Make sure I rendered your question correctly. Also would be nice if you use it as a small template for future questions in terms of latexifying. $\endgroup$ – Kaster May 29 '14 at 23:56
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Presumably $X$ is uniformly distributed over $(0,2 \pi)$.

Hint: Note that $\sin x$ takes on the same value at $x$ and $\pi - x$.

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