Challenging +1.8 This question requires deriving a CDF through transformation of a continuous random variable, involving inverse trigonometric functions and careful consideration of the sine function's non-monotonic behavior over [-π, π]. It demands understanding of the relationship between PDFs/CDFs under transformations and handling the multi-valued inverse, which goes beyond routine application of formulas.
9.
The continuous random variable \(X\) has a uniform distribution on the interval \([ - \pi , \pi ]\).
The random variable \(Y\) is defined by \(Y = \sin X\).
Determine the cumulative distribution function of \(Y\).
END OF TEST
9.
The continuous random variable $X$ has a uniform distribution on the interval $[ - \pi , \pi ]$.\\
The random variable $Y$ is defined by $Y = \sin X$.
Determine the cumulative distribution function of $Y$.
END OF TEST
\hfill \mbox{\textit{SPS SPS FM Statistics 2024 Q9 [7]}}