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Homework Help: Function of random variables

  1. Jul 25, 2014 #1
    Suppose X ~ U[ 0, pi ]

    What is the distribution of Y=sinX.

    I have a solution in my notes however I don,t understand the following the second transition:
    F_Y(y) = P(Y \leq y) = P(X \leq \arcsin(y)) + P(X \geq \pi - \arcsin(y)) = ...

    Where the [itex] P(X \geq \pi - \arcsin(y)) [/itex] comes from?
  2. jcsd
  3. Jul 25, 2014 #2


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    The sine function is not monotonic.

    Y is smaller than y if either of the arguments on the RHS are fulfilled.
  4. Jul 25, 2014 #3
    Since the sin function is symmetric about ∏/2, there are two possible ranges you need to consider for the same values(imagine drawing a straight horizontal line at any 'y' on the sin curve, there are two symmetric value intervals below it), from [0, arcsin(y)] and [∏ - arcsin(y), ∏] which can be reflected in the solution.
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