Let $\underline{X}=(X_1,...X_n)$ be an i.i.d. random sample from an exponential distribution, with probability density function given by
$f(x; \theta)=\lambda exp${$-\lambda x$} , x>0 where $\lambda$ is an unknown parameter taking values in $\mathbb{R^+}$
A Derive the likelihood function $L(\lambda; \underline{X})$ and derive the Fisher information $I(\lambda)$ measuring the amount of information that $\underline{X}$ carries about $\lambda$
B Given the Jeffreys prior for $\lambda$ is $\pi_J(\lambda)\propto \lambda^{-1}$, derive the posterior distribution for $\lambda$. Find the mean and variance of $\lambda| \underline{X}$
I know that for A I need to use the product up to n and then for the fisher info it is $\frac{1}{-E(l''(\lambda)}$ but I'm not sure how to do this for my given model.
A: the likelihood is the following:
$$L(\lambda)=\lambda^n e^{-\lambda\Sigma_i X_i}$$
To calculate Fischer information, as you stated, you have to calculate
$$-n\mathbb{E}\Bigg[\frac{\partial^2}{\partial\lambda^2}\log f(x;\lambda)\Bigg]$$
thus simply:
$$\log f=\log\lambda-\lambda x$$
$$\frac{\partial^2}{\partial\lambda^2}\log f=-\frac{1}{\lambda^2}$$
So the Fischer information of the n-tuple $(X_1\dots,X_n)$ is
$$\frac{n}{\lambda^2}$$
B
The posterior is the following
$$\pi(\lambda|\mathbf{x})\propto \pi(\lambda)\cdot p(\mathbf{x}|\lambda)$$
that is
$$\pi(\lambda|\mathbf{x})\propto \lambda^{-1}\times \lambda^n e^{-\lambda\Sigma_i X_i}= \lambda^{n-1}e^{-\lambda\Sigma_i X_i}$$
We immediately recongize the kernel of a $Gamma(n;\Sigma_i X_i)$ thus
$$\mathbb{E}[\lambda|\mathbf{x}]=\frac{n}{\Sigma_i X_i}$$
and
$$\mathbb{V}[\lambda|\mathbf{x}]=\frac{n}{(\Sigma_i X_i)^2}$$