Here we continue the treatment of continuous probability functions (distributions) begun in Probability Functions: Part 2 and Continuous distributions. Our emphasis is on distributions derived from the normal distribution. We begin with the Chi-square distribution, which was already considered in Further Continuous Distributions. But here we approach it as a sampling distribution. We then consider with two further sampling distributions; the **Student’s t **and **F distributions**.

The first distribution we consider is closely tied to the notion of **sample** introduced in Statistics Defined and extended in Samples and Sampling. In particular, consider a random sample obtained from a population for which the underlying probability

function is known. It is then possible to calculate various sample statistics. From these we can derive one or more distributions related to the sample, called **sampling distributions**. A sampling distribution may be described as …

One sampling distribution important in the hypothesis-testing of Inferential Statistics is the Chi-square distribution.

The graphs of two instances of the continuous probability function known as the **chi-square distribution** are shown below.

Note

the asymmetry of the two functions. That is, most of the probability sits below the mean/df … we say it is skewed right. Note also that these are density functions [pdf’s] of

the continuous random variable X. As such, they do not output

probabilities. As discussed in Probability Functions: Part 2

it is the cumulative distribution function, F(X), of a continuous

random variable which outputs a probability. The ‘TI-83 plus’ function

which outputs probabilities related to the Chi-square distribution is **cdf**.

Just as there are different normal distributions corresponding to different

population means and standard deviations, so there are different

chi-square distributions. The population parameter which varies in the

above distributions is called the **degrees of freedom, df**. This

quantity may be defined as a constant (or parameter) based on the

number of sample classes, k. In particular, if expected frequencies can

be computed without having to estimate population parameters from

sample statistics, then df = k – 1. The degrees of freedom parameter is

sufficient to define a chi-square probability density function or pdf.

In one of the above distributions, df = 7; in the other, df = 2.

Although the chi-square pdf involves **post-secondary mathematics**,^{*}

graphics calculators currently (2007) used in secondary classrooms are

quite capable of generating the function for various degrees of freedom

(values of df). Therefore, a reader interested in applications of the

chi-square distribution may find a number of examples obtained with the

TI-83 plus calculator in Inferential Statistics.

A second important sampling distribution is the **Student’s t-distribution**, or t-distribution for short. The graphs of two instances of this continuous probability function are shown below.

Note

symmetry. That is, each function takes on the same values on either

side of the line t = 0. Note further that these are pdf’s..

…

Another important sampling distribution is the so-called **Student’s t distribution**.

As in

the chi-square distribution, there are different t-distributions

corresponding to different degrees of freedom, df. In the example

above, df = 1 and 5.

Although the general expression for the Student’s t density function involves post-secondary mathematics^{**},

the function is once again accessible via graphics calculators commonly

used by secondary students. To illustrate this point, a number of

t-distribution applications obtained with the ‘TI-83 plus’ calculator

are presented in Inferential Statistics.

A third important sampling distribution is the **F-distribution**.

As for the two sampling distributions considered above, there are

different F -distributions corresponding to different degrees of

freedom. For this distribution, however, there are two degrees of

freedom parameters, m and n. A graph of the F-distribution for m = 10

and n = 15 is shown below.

..

Note asymmetry…

The

general expression for the F-distribution density function is even more

complex than those of the chi-square and t-distributions, but is once

again accessible to secondary students via modern graphics calculator

technology. A number of F-distribution applications are explored using

the TI-83 plus calculator in Inferential Statistics.

To complete our coverage of continuous probability functions we now turn briefly to the function known as the **beta distribution**.

Like the sampling distributions we have considered above, the exp., gamma & chi-sq

functions are related to the normal distribution. We therefore conclude

this essay with a demonstration of the connections between the normal

distribution and the six other continuous distributions we have covered.

conclusion…

* f(x) =

**

conclusion…

* f(x) =

**

Our emphasis is on distributions which are functions of two or more random variables. .. ? (sampling distribution)

We conclude by examining a continuous distribution important in Bayesian statistics; the beta distribution.