Question 1 assume we have an iid sample from bernoulli

Question

Question 1: Assume we have an i.i.d. sample from Bernoulli random variable, specifically X₁, X₂... ,X_n ~ B (p), which means

f (X_i) = p^xi (1 - p)^{1 - xi}, X_i = 0, 1.

In class we showed that

Is a stationary point for the likelihood for the parameter p. Show p indeed maximizes the likelihood?

(Hint: find the second derivative d²/dp² log L(p¦X₁, ... , X_n), and show that it is negative at p)

Problem 2: Assume the random variable X follows the exponential distribution through the following parameterization of its probability density function

f (x) = Θe -Θx, for x ≥ 0.

If you have and i.i.d sample X₁, X₂, ... , X_n from this distribution, what is maximum likelihood estimator for Θ?