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

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

X1234
Px(x)θp(1-θ)pθ(1-p)(1-θ)(1-p)
Table 1: Pmf of X

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:

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