Bayesian updating for a wait-time rate, the Exponential-Gamma

Question

You model the time between trades as Xi \sim 	ext{Exponential}(\lambda) with unknown rate \lambda, and place a Gamma prior \lambda \sim 	ext{Gamma}(\alpha, \beta) (shape \alpha, rate \beta) with \alpha = 2, \beta = 1. You then observe n = 3 inter-trade times summing to 2 hours. Derive the posterior, give its mean, and interpret the prior's weight.

Accepted Answer

Likelihood. The Exponential density is \lambda e^{-\lambda x}, so for n independent gaps with total time S = \sumi xi, p(	ext{data}\mid\lambda) = \prod{i=1}^n \lambda e^{-\lambda xi} = \lambda^{n} e^{-\lambda S}. Prior. The Gamma(\alpha,\beta) density is p(\lambda) \propto \lambda^{\alpha-1} e^{-\beta\lambda}. Posterior. Multiply and collect powers of \lambda: p(\lambda\mid	ext{data}) \ \propto\ \lambda^{\alpha-1}e^{-\beta\lambda}\cdot \lambda^{n}e^{-\lambda S} = \lambda^{\,\alpha+n-1}\,e^{-(\beta+S)\lambda}. That is a Gamma kernel, so \lambda \mid 	ext{data} \ \sim\ 	ext{Gamma}(\alpha + …

Bayesian updating for a wait-time rate, the Exponential-Gamma

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