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Derivative of cumulative function

  1. Aug 11, 2009 #1
    Hi everybody,

    Can someone tell me the formula to I should use to find the derivative of the following function, with respect to x:

    F(x)=Probability[Y<=g(x)]

    dF(x)/dx = ??

    Thank you for your help.

    Toltol
     
    Last edited: Aug 11, 2009
  2. jcsd
  3. Aug 11, 2009 #2
    Well, first step, let h(y) be the density function for Y, and let H(y) be the cumulative distribution function for Y. Now we have
    F(x)=P(Y<=g(x))
    =[tex]\int_{-\infty}^{g(x)} h(y) dy[/tex]
    =H(g(x))
    Now, can you differentiate that?
     
  4. Aug 12, 2009 #3
    Thank you mXSCNT.

    If F(x)=P[Y<=g(x)]=H[g(x)]

    Thus, the derivative is:

    F'(x)=dH[g(x)]/dg(x) . dg(x)/x

    The term dH[g(x)]/dg(x) is >0; Thus, the sign of F'(x) is the sign of dg(x)/x.

    Am I ok?

    Thank you,
    Toltol
     
  5. Aug 12, 2009 #4
    H'(g(x)) g'(x) = h(g(x)) g'(x)
     
  6. Aug 12, 2009 #5
    Thank you mXSCNT.

    It's OK.
     
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