Question: The likelihood function of \(\mathrm{n}\) random variables \(\mathrm{X}_{1}, \mathrm{X}_{2}, \ldots \mathrm{X}_{n}\) is defined to be the joint density of \(\mathrm{n}\) random variables \(\mathrm{f}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \mathrm{x}_{\mathrm{m}}, \theta\right)\), which is considered to be a function of one or more parameters \(\bar{\theta}=\left(\theta_{1}, \theta_{2}, \ldots, \theta_{n}\right)\). The likelihood function is defined by

\(\mathrm{L}\left(\theta, \mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \mathrm{x}_{\mathrm{n}}\right)=\mathrm{f}\left(\mathrm{x}_{1} ; \theta\right) \mathrm{f}\left(\mathrm{x}_{2} ; \theta\right) \ldots \mathrm{f}\left(\mathrm{x}_{\mathrm{n}} ; \theta\right)\)

Assume that a random sample of size \(\mathrm{n}\) from the normal distribution has the density function

where \(\mu\) and \(\sigma^{2}\) are the mean and variance for each \(\mathrm{x}_{\mathrm{i}}\) (note the \(\mathrm{x}_{\mathrm{i}} \mathrm{s}\) are independent).

1. If \(\bar{\theta}=\left(\theta_{1}, \theta_{2}\right)=\left(u, \sigma^{2}\right)\) is the parameter space for the density function \(\mathrm{f}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots .\right.\)
   \(\mathrm{x}_{\mathrm{n}}\) ) then what is the likelihood function?
2. Find the maximum likelihood estimator \(\theta_{1}\) of \(\theta_{1}\) which maximizes the likelihood function. (Hint: Take the log of the likelihood function, then set its derivative with
   respect to \(\theta_{1}\) equal to zero and solve for \(\theta_{1 .}\) )
3. Find the maximum likelihood estimator \(\theta_{2}\) of \(\theta_{2}\) which maximizes the likelihood function. (Same as b except for \(\theta_{2}\) )

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