<Dd> G a m m a (λ; α, β) = β α Γ (α) λ α − 1 exp ⁡ (− λ β). (\ displaystyle \ mathrm (Gamma) (\ lambda; \ alpha, \ beta) = (\ frac (\ beta ^ (\ alpha)) (\ Gamma (\ alpha))) \ lambda ^ (\ alpha - 1) \ exp (- \ lambda \ beta).) </Dd> <P> The posterior distribution p can then be expressed in terms of the likelihood function defined above and a gamma prior: </P> <Dl> <Dd> p (λ) ∝ L (λ) × G a m m a (λ; α, β) = λ n exp ⁡ (− λ n x _̄) × β α Γ (α) λ α − 1 exp ⁡ (− λ β) ∝ λ (α + n) − 1 exp ⁡ (− λ (β + n x _̄)). (\ displaystyle (\ begin (aligned) p (\ lambda) & \ propto L (\ lambda) \ times \ mathrm (Gamma) (\ lambda; \ alpha, \ beta) \ \ & = \ lambda ^ (n) \ exp \ left (- \ lambda n (\ overline (x)) \ right) \ times (\ frac (\ beta ^ (\ alpha)) (\ Gamma (\ alpha))) \ lambda ^ (\ alpha - 1) \ exp (- \ lambda \ beta) \ \ & \ propto \ lambda ^ ((\ alpha + n) - 1) \ exp (- \ lambda \ left (\ beta + n (\ overline (x)) \ right)). \ end (aligned))) </Dd> </Dl> <Dd> p (λ) ∝ L (λ) × G a m m a (λ; α, β) = λ n exp ⁡ (− λ n x _̄) × β α Γ (α) λ α − 1 exp ⁡ (− λ β) ∝ λ (α + n) − 1 exp ⁡ (− λ (β + n x _̄)). (\ displaystyle (\ begin (aligned) p (\ lambda) & \ propto L (\ lambda) \ times \ mathrm (Gamma) (\ lambda; \ alpha, \ beta) \ \ & = \ lambda ^ (n) \ exp \ left (- \ lambda n (\ overline (x)) \ right) \ times (\ frac (\ beta ^ (\ alpha)) (\ Gamma (\ alpha))) \ lambda ^ (\ alpha - 1) \ exp (- \ lambda \ beta) \ \ & \ propto \ lambda ^ ((\ alpha + n) - 1) \ exp (- \ lambda \ left (\ beta + n (\ overline (x)) \ right)). \ end (aligned))) </Dd>

Distribution of the sum of exponential random variables