Integral of Weibull distribution
By John Parsons •
How would you find the distribution function for the following density functions (Weibull function):
$$f_{X}(x) = c\tau x^{\tau−1}e^{− cx^{\tau}} $$
for $0< x < \infty$, $\tau > 0$ and $c>0$.
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$\begingroup$So $$F_X(x)=\int_0^x c\tau s^{\tau−1}e^{− cs^{\tau}} ds\\ = c\tau \int_0^x s^{\tau−1}e^{− cs^{\tau}} ds $$
If we use the substitution $s^{\tau}=u$, and $\frac{du}{ds}=\tau s^{\tau-1}$ this simplifies to
$$c\int_0^{x^\tau} e^{− cu} du\\ =\left[-e^{-cu}\right]_0^{x^\tau}\\ =1-e^{-cx^{\tau}}.$$
I hope that I've not given this to you too easily and that this is useful to you.
$\textbf{EDIT}$: I have assumed you were asking for the c.d.f. but the other commenters are correct your question is not entirely clear on its terminology. Also fixed my $\LaTeX$.
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