# 1.1 - Definitions

1.1 - Definitions

We'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.

 [1] Link ↥ Has Tooltip/Popover Toggleable Visibility