I estimated the shape $\alpha$ and scale $\lambda$ parameter of the Weibull distribution using Bayesian methods. That gave me a marginal posterior distributions for both parameters. Calculating the highest posterior density (HPD) I got two credibility intervals and a point estimate for both paramters.
Using the point estimates I can calculate the probability that a Weibull distributed variable $X$ will yield a value smaller than $x$.
$F(x) = 1 - exp(- (x/\lambda)^\alpha)$
Now I also want to create a credibility interval for that probability.
How can I calculate the credibility interval for $F(x)$?
Lets say that using the HPD I got credibility intervals for $\alpha$ and $\lambda$ that are [0.52 ; 0.74] and [156.3 ; 187.8] respectively. The point estomates are 0.61 and 175.5.
Using the point estimates and $x = 25$ one gets $F(25) = 0.262584$
What I did so far:
I used the credibility intervals of the parameters and calculated $F(25)$ for all possible combinations of the interval values and point estimates.
- 0.52 & 156.3 $\rightarrow$ $F(25) = 0.3199171$
- 0.52 & 175.5 $\rightarrow$ $F(25) = 0.3044124$
- 0.52 & 178.8 $\rightarrow$ $F(25) = 0.3019741$
- 0.74 & 178.8 $\rightarrow$ $F(25) = 0.2080028$
I then used the minimum of these values as the lower probability interval and the maximum as the upper probability interval. But I don't know if that is a valid thing to do.