# 1.1 - Definitions

1.1 - DefinitionsWe'll start the lesson with some formal definitions. In doing so, recall that we denote the \(n\) random variables arising from a random sample as subscripted uppercase letters:

\(X_1, X_2, \cdots, X_n\)

The corresponding observed values of a specific random sample are then denoted as subscripted lowercase letters:

\(x_1, x_2, \cdots, x_n\)

- Parameter Space
- The range of possible values of the parameter \(\theta\) is called the
**parameter space**\(\Omega\) (the greek letter "omega").

For example, if \(\mu\) denotes the mean grade point average of all college students, then the parameter space (assuming a 4-point grading scale) is:

\(\Omega=\{\mu: 0\le \mu\le 4\}\)

And, if \(p\) denotes the proportion of students who smoke cigarettes, then the parameter space is:

\(\Omega=\{p:0\le p\le 1\}\)

- Point Estimator
- The function of \(X_1, X_2, \cdots, X_n\), that is, the statistic \(u=(X_1, X_2, \cdots, X_n)\), used to estimate \(\theta\) is called a
**point estimator**of \(\theta\).

For example, the function:

\(\bar{X}=\dfrac{1}{n}\sum\limits_{i=1}^n X_i\)

is a point estimator of the population mean \(\mu\). The function:

\(\hat{p}=\dfrac{1}{n}\sum\limits_{i=1}^n X_i\)

(where \(X_i=0\text{ or }1)\) is a point estimator of the population proportion \(p\). And, the function:

\(S^2=\dfrac{1}{n-1}\sum\limits_{i=1}^n (X_i-\bar{X})^2\)

is a point estimator of the population variance \(\sigma^2\).

- Point Estimate
- The function \(u(x_1, x_2, \cdots, x_n)\) computed from a set of data is an observed
**point estimate**of \(\theta\).

For example, if \(x_i\) are the observed grade point averages of a sample of 88 students, then:

\(\bar{x}=\dfrac{1}{88}\sum\limits_{i=1}^{88} x_i=3.12\)

is a point estimate of \(\mu\), the mean grade point average of all the students in the population.

And, if \(x_i=0\) if a student has no tattoo, and \(x_i=1\) if a student has a tattoo, then:

\(\hat{p}=0.11\)

is a point estimate of \(p\), the proportion of all students in the population who have a tattoo.

Now, with the above definitions aside, let's go learn about the method of maximum likelihood.