Derivative of cumulative function

[toltol]

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

 

[mXSCNT]

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))
=$$\int_{-\infty}^{g(x)} h(y) dy$$
=H(g(x))
Now, can you differentiate that?

 

[toltol]

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

 

[mXSCNT]

H'(g(x)) g'(x) = h(g(x)) g'(x)

 

[toltol]

Thank you mXSCNT.

It's OK.