**Question: Discrete Example Suppose we have a 4-sided die and let X denote the random face that comes up on a throw. Its pmf is given by Table 1, where θ,p ∈ [0,1]. Suppose we throw**

X | 1 | 2 | 3 | 4 |

Px(x) | θp | (1-θ)p | θ(1-p) | (1-θ)(1-p) |

the die a certain number of times and observe xi i’s, for i = 1,…,4 (i.e., face i comes up xi times).

(a) What is the likelihood of this experiment given θ? (You should treat p as a constant) (b) What is the maximum likelihood estimate of θ?

3.2 Normal Distribution

Suppose we sample n i.i.d. points from a Gaussian distribution with mean μ and variance σ?. Recall that the Gaussian pdf is given by

Compute the maximum likelihood estimate of parameters μ and σ?.

**Solution: **