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Non 1-1 transformation of continuous random variable

  1. Feb 2, 2010 #1
    1. The problem statement, all variables and given/known data

    X is exponentially distributed with mean s.
    Find P(Sin(X)> 1/2)

    2. Relevant equations

    fX(x) = se-sx, x[tex]\geq[/tex] 0
    0, otherwise

    FX(x) = 1 - e-sx, x[tex]\geq[/tex] 0
    0 otherwise

    3. The attempt at a solution

    Let Y = sin X

    FY (y) = P(Y[tex]\leq[/tex] y)
    = P(sinX [tex]\leq[/tex] Y)
    = P(X [tex]\leq[/tex] arcsin(y), X[tex]\geq[/tex] [tex]\pi[/tex] - arcsin(y)) {This is where I become slightly unsure}
    =FX(arcsin(y)) - FX([tex]\pi[/tex] - arcsin(y))
    =1-e-s(arcsin(y)) - (1-e-s([tex]\pi[/tex] - arcsin(y)))

    From here I can differentiate to find the pdf and then use that to find sinX < 1/2.
  2. jcsd
  3. Feb 4, 2010 #2
    Any ideas anyone?
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