Python command

Likelihood function is an important concept in statistical inference, it is defined as the joint probability (for discrete random variable) or joint probability density (for continuous random variable) of the sample. For example, if we collect a random sample 1,2,, from normal distribution with mean and standard deviation , then the likelihood function is given by (,;)==(1/2 ) e^ -()/2^2

First, let's write a function to compute the value of the likelihood function for a normal random sample.

x = [8.0, 8.3, 5.5, 4.2, 3.3, 3.0, 5.5, 2.9, 4.9, 4.2, 3.2, 1.6, 5.4, 0.3, 3.5, 9.4, 6.7, 7.3, 5.6, 4.9]

2. Let's use this function to obtain the maximum likelihood estimate. Namely, we need to find the values for parameters and such that the likelihood function of the random sample is maximized. It can be shown that likelihood function and its natural logarithm (known as log-likelihood function) achieve the maximum at the same (,) pair, we will use negative log-likelihood function as our objective function.

Use the minimize function to find the maximum likelihood estimate of and for a normal random sample.