Hi, I am struggling a bit on this question. For part A of Jeffrey's prior, I found that it was . Is this correct? I did this by taking the log likelihood and finding its second derivative. Next, I square rooted. I'm more so confused on where to begin for parts b - d. Any help would be appreciated.

Suppose x1, . . . , xn is a random sample from an exponential distribution with mean 1/.

a) Derive the Jeffrey's prior for

b) Derive the posterior distribution of using the Jeffrey's prior.

c) Derive the predictive distribution of a future observation z.

d) Derive a 95% credible interval for z.