I'm having trouble with the following problem:
Suppose a machine is composed of 2 components (1 and 2, independent from each other). Each component has a exponential failure probability distribution with $\lambda$ parameter. There are 10 of these machines. The time of failure of these 10 machines is observed (X is the failiure time for component 1 and Y is the failure time for 2). Suppose a Gamma prior distribution for $\lambda$ with parameter $a$ and $b$. What is the posterior distribution of $\lambda$. Test the hypotheses that $\lambda > t$ versus $\lambda < t$ in a Bayesian way.
My answer until now: $Z = X + Y, X \sim exp(\lambda), Y \sim exp(\lambda), f_{Z}(z) = \lambda^{2}ze^{-\lambda z} $.
So $\lambda |z \sim Gamma(20+a,\sum_{i=1}^{10}z_{i}+b)$.
How can I do a hypothesis test in a Bayesian way with this posterior?
You calculate the probability of the 2 hypothesis and then propose $P(Ho)>P(H1)$ and see under what conditions this statement is true .In this case:
$P(H_0)=P(λ>t)=1-P(λ<t)=1-F_λ(t)$ you already know the posteriori distribution ,just integrate respect to λ and $P(H_1)=1-P(H_0) $