My math tutor gave me a set of problems, including the following that asks to show the steps taken to integrate to find the CDF:
$$f(x) = \frac{k}{λ}\left(\frac{x}{λ}\right)^{(k-1)}e^{-\left({x/λ}\right)^k}$$
Please can one of you guys very kindly explain how I work through step by step so as I can understand how it happens and so I might continue with the other work set?
Thanks in advance
Weibull PDF to CDF help needed please!
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$\begingroup$
calculus
integration
probability-distributions
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0I have spent a long while trying to formulate an answer, but I'm getting no-where, I really don't get it, that's why I've asked for help. – 2017-01-18
1 Answers
1
Hint:
The support of the distribution is the set of nonnegative value, for $x \geq 0$, you have to evaluate
$$\int_0^{x} f(t) dt=\int_0^{x} \frac{k}{\lambda}\left( \frac{t}{\lambda} \right)^{k-1} \exp\left(-\left( \frac{t}{\lambda}\right)^k\right) dt$$
Notice that $$\frac{d}{dt}\exp\left( -\left( \frac{t}{\lambda}\right)^k\right)=-\exp\left( -\left(\frac{t}{\lambda} \right)^k \right)\frac{k}{\lambda}\left( \frac{t}{\lambda}\right)^{k-1}$$