2. This problem copes with a new distribution: exponential distribution.

Let x = {(²) € R}/₁ be an i.i.d. sample of N univariate instances drawn from an

exponential density function:

p(z(0) | 0) = { 0exp(-0z(0) x(0) 20

(e) <0,

where > 0 is called the rate parameter of the distribution.

(a) Find a point estimate of using maximum likelihood estimation. Show all the steps

in your derivation.

(b) Suppose is now treated as a random variable (instead of a fixed parameter as above)

with prior density p(0) of the following form:

p(0) x 0-¹ exp(-03),

where a, 3 > 0 are two parameters.

Show that the posterior density p(0 | X) has the same form as p(0) but with different

values for a and B. What are the new parameter values?

(a) 10 points

(b) 10 points