Further Continuous Distributions: Part 2

chi_sq_2

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.

Student's t Distributions with df = 5 and 1

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.

Student's t Distributions with df = 5 and 1

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.

Student's t Distributions with df = 5 and 1

..

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.