# Exponential distribution and gamma prior

I want to use Bayesian conjugate to update my prior. Let's say I model bus arrivals by Exponential distribution with lamba=0.5. It means on average I will wait for 2h = 1/0.5

Prior is gamma with: alha=1, beta=2, E[gamma] = alha/beta = 1/2

I have observed: 'the bus arrived after 3 hours'. Posterior: alha += 1, beta +=3, E[gamma] = 2/5 My posterior expected waiting time will be longer 5/2=2.5

How should I update my prior in case I have observed the next event: 'I waited for 3h and went home'. There is no information when the 'bus has arrived' but that 'it did not arrive for 3h'

The prior distribution on $$\lambda$$ is the Gamma prior $$\pi(\lambda)\propto\lambda^{\alpha-1}\exp\{-\beta\lambda\}\,\mathbb I_{\lambda>0}$$ The observation is the non-realisation of an event $$E$$. The probability that it did not occur is $$\mathbb P_\lambda(E^c)$$, which is also the likelihood. Hence the posterior writes as $$\pi(\lambda|E^c)\propto\pi(\lambda)\times\mathbb P_\lambda(E^c)$$ Since the event $$E^c$$ means that the bus took more than three hours to come $$\mathbb P_\lambda(E^c)=\mathbb P_\lambda(X>3)=\int_3^{\infty} \lambda e^{-\lambda x}\,\text dx=e^{-3\lambda}$$ Therefore $$\pi(\lambda|E^c)\propto \lambda^{\alpha-1}\exp\{-\beta\lambda\} \times \exp\{-3\lambda\}\,\mathbb I_{\lambda>0}$$
• It is also nice to notice that the posterior is Gamma with parameters $\alpha,\beta+3$ Commented Nov 29, 2020 at 6:37