Final answer:
To find the joint posterior distribution of Q1 and Q2, use Bayesian inference with the given data and the prior distribution. The median and credible intervals for the prior and posterior distributions are found using the properties of the inverse-gamma distribution.
Step-by-step explanation:
The problem involves determining the joint posterior distribution of Q1 and Q2 for the old and new manufacturing processes, which are modeled as exponential random variables. Given that the prior distribution for Q1 and Q2 is an inverse-gamma distribution with shape 1 and inverse-scale 1/30, we would update the parameters of the prior distribution with the data from the stress tests to get the posterior distribution. Calculating the posterior distribution requires Bayesian inference, which involves using the likelihood function derived from the exponential probability density function and the prior distribution to determine the new shape and scale parameters. Once the posterior distribution parameters are found, you can calculate the median and 95% credible interval from the inverse-gamma distribution's properties.
For prior distributions: An inverse-gamma distribution with shape parameter α and scale parameter β has mean β / (α - 1) for α > 1 and variance β² / [(α - 1)²(α - 2)] for α > 2. Therefore, for a shape of 1 and inverse-scale 1/30, the mean and variance do not exist as these formulas are not defined for α ≤ 1.
For posterior distributions: After updating the prior with the data, the shape parameter α' becomes α + n (the number of observations), and the scale parameter β' becomes β + Σx (the sum of the observed failure times). The median of the inverse-gamma distribution can be found using statistical software, as it involves inverting the cumulative distribution function, and credible intervals could be found similarly or by using a formula specific to the inverse-gamma distribution.