Suppose $Y_1, \dots Y_n$ are exponentially distributed: $Y_i | \lambda \sim Exp(\lambda)$. Find the conjugate prior for $\lambda$, and the corresponding posterior distribution. Show that the posterior mean for the failure rate $\lambda$ can be written as a weighted average of the prior mean and the maximum likelihood estimator, $\hat{\lambda}=\bar{y}^{-1}$.
Given that the joint pdf as follow:-
$P(Y_i | \lambda)=\lambda^{n}e^{-\lambda \sum_{i=1}^{n} Y_i}$
Then the likelihood function is
$ L (\lambda | Y_i)\propto \lambda^{n}e^{-\lambda \sum_{i=1}^{n}Y_i}$
which give Gamma distribution with $\alpha = n+1$ and $\beta =\sum_{i=1}^{n} Y_i$.
Thus, with the prior density
$P(\lambda)\propto \lambda^{a-1} e^{- \lambda b}$
gives the posterior distribution as follow:-
$P(\lambda | Y_i) \propto \lambda^{n+a-1} e^{-\lambda (\sum_{i=1}^{n}Y_i + b)}$
and resulting in Gamma distribution with $\alpha = n+a$ and $\beta = \sum_{i=1}^{n} Y_i + b$.
My problem now is how can I show the posterior mean for the failure rate λ can be written as a weighted average of the prior mean and the maximum likelihood estimator?
As far I get as the following:-
$E[\lambda | Y] = \frac{\alpha}{\beta} = \frac{n+a}{\sum_{i=1}^{n} Y_i + b}$
Can anyone help me to show how the posterior mean above can be written as a weighted average of the prior mean and the maximum likelihood estimator, $\hat{\lambda}=\bar{y}^{-1}$?