The lifetime (in thousands of hours) X of a light bulb has pdf
g(x)= \lambda e^{-\lambda x}, \hspace{3mm} x\geq 0
for some unknown \lambda >0.
We collect {\color{blue}{n=33}} independent lightbulbs at random and record their lifetime X_1,\ldots ,X_ n, which are all independent copies of X. We find that {\color{blue}{\overline{X_ n}=42.6}} thousand hours.
Now, consider the prior distribution \lambda \sim \mathsf{Exp}(\theta ), for some fixed parameter \theta >0.
Compute the posterior pdf \displaystyle \pi (\lambda |X_1,\ldots , X_ n) of \lambda up to normalizing constant. Enter your answer in terms of \lambda, \theta, n and S_ n=\sum _{i=1}^{n} X_ i.
(Enter S_n for S_ n=\sum _{i=1}^{n} X_ i. )
(Any answer correct up to a normalizing constant will be accepted.)
\pi (\lambda |X_1,\ldots , X_ n)=\quad
Compute the maximum a posteriori estimator \hat{\lambda }^{\text {MAP}} of \lambda.
Enter your answer in terms of \theta, n and S_ n=\sum _{i=1}^{n} X_ i.
(Enter S_n for S_ n=\sum _{i=1}^{n} X_ i. )
\hat{\lambda }^{\text {MAP}}=\quad