A bakers oven may be out of use due for two reasons. With probability 0.8 the oven will be damaged from dirt and it will take exactly 5 minutes to repair it. With probability 0.2 the oven will need major repairs and repair time will follow a Weibull distribution with parameters with α = 6 and β = 0.5.
a) If X is the repair time of the next failure, find the cumulative distribution of X. b) Outline an inversion method to generate the failure times.
I am confused with the above problem for a I tried to generate the cdf by using the weibull cdf and adding the probabilities and arrived at:
$F(x)=0$ for $x<1-e^-(\frac{x}{6})^{0.5}$
$F(x)=0.2$ for $1-e^-(\frac{x}{6})^{0.5}<x<5$
$F(x)=1$ for $5<x$
But I am not sure if this is right and then I also am not sure how to use the inversion method on a 3 tier function. Any help would be appreciated!
Correction: the CDF is $F=1-\exp\sqrt{x/6}$ for $x\ge 0$ and $0$ otherwise. (Look up the usual Weibull CDF wherever you prefer, or integrate it's pdf.)I don't know how you got your answer, but any piecewise formula for the CDF add a function of $x$ shouldn't transition at $x$-dependent values. I'll leave you to express $x$ as a function of $F$.