Now, suppose that we instead have the proper prior \pi (\lambda ) \sim \textsf{Exp}(a) (a > 0). Again, just as in part (b): conditional on \lambda, we have observations X _1, X _2, \cdots, X _{n} \stackrel{\text {i.i.d}}{\sim } \textsf{N}(\lambda , 1). You may assume that a < \displaystyle \sum _{i=1}^{n} X_ i. Compute the posterior distribution \pi (\lambda | X_1, X_2, \ldots , X_ n), then provide the following statistics on the posterior distribution. Write Phi for the CDF function \Phi () and PhiInv for its inverse.

Use SumXi for \sum _{i=1}^ n X_ i.

median:

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To compute the posterior distribution, we use Bayes' theorem:

\(\pi(\lambda|X_1,X_2,...,X_n) \propto \pi(\lambda) \times \prod_{i=1}^{n} f(X_i|\lambda)\)

where \(\pi(\lambda)\) is the prior distribution and \(f(X_i|\lambda)\) is the likelihood function.

In this case, the prior distribution \(\pi(\lambda)\) is an exponential distribution, given by \(\pi(\lambda) \sim \textsf{Exp}(a)\).

The likelihood function is given by \(f(X_i|\lambda) = \frac{1}{\sqrt{2\pi}}e^{-\frac{(X_i-\lambda)^2}{2}}\).

Therefore, the posterior distribution is given by:

\(\pi(\lambda|X_1,X_2,...,X_n) \propto e^{-a\lambda} \times e^{-\frac{1}{2}\sum_{i=1}^{n}(X_i-\lambda)^2}\)

Multiplying the exponential terms and simplifying, we get:

\(\pi(\lambda|X_1,X_2,...,X_n) \propto e^{-(a+\frac{n}{2})\lambda + \sum_{i=1}^{n}X_i\lambda - \frac{1}{2}\sum_{i=1}^{n}X_i^2}\)

We recognize the exponential distribution in the exponent, so we can factor out the normalizing constant to find the proportional form of the posterior distribution:

\(\pi(\lambda|X_1,X_2,...,X_n) \propto e^{-(a+\frac{n}{2})\lambda}\times e^{-\frac{1}{2}(\sum_{i=1}^{n}X_i^2 - 2\lambda\sum_{i=1}^{n}X_i)}\)

This is a proportional form of a gamma distribution with shape parameter \(a^* = a + \frac{n}{2}\) and rate parameter \(b^* = \sum_{i=1}^{n}X_i\).

Therefore, the posterior distribution is given by \(\pi(\lambda|X_1,X_2,...,X_n) \sim \textsf{Gamma}(a^*, b^*)\).

Now, let's move on to computing the median and mode of the posterior distribution.

The median of a gamma distribution is given by \(\lambda = \frac{a^* - 1}{b^*}\). Therefore, the median of the posterior distribution is:

\(\text{median} = \frac{a + \frac{n}{2} - 1}{\sum_{i=1}^{n}X_i}\)

The mode of a gamma distribution is given by \(\lambda = \frac{a^* - 1}{b^*}\) if \(a^* > 1\), otherwise the mode is undefined. Therefore, the mode of the posterior distribution is:

\(\text{mode} = \frac{a + \frac{n}{2} - 1}{\sum_{i=1}^{n}X_i}\) if \(a + \frac{n}{2} > 1\), otherwise the mode is undefined.