# Derivative of cumulative function

1. Aug 11, 2009

### 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 = ??

Toltol

Last edited: Aug 11, 2009
2. Aug 11, 2009

### 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?

3. Aug 12, 2009

### 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

4. Aug 12, 2009

### mXSCNT

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

5. Aug 12, 2009

### toltol

Thank you mXSCNT.

It's OK.