MATH 11400 Statistics 1 2008-09
6. Exact sampling distributions related to the Normal distribution
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Aims
Knowing the exact distribution of an estimator helps us to understand how its behaviour
depends, for example, on the sample size or the unknown population parameter values.
It also enables us to incorporate our theoretical results into other
aspects of our statistical analysis.
In this section we derive the exact distribution
of some sample statistics associated with random samples from
a range of distributions, particularly
focussing on the mean and variance of a random sample from the Normal distribution.
Objectives
The following objectives will help you to assess how well you have mastered the relevant material.
By the end of this section you should be able to:
Recall the distibution of the sample mean
for a simple random sample from a Normal distribution
and and use statistical tables
or an appropriate statistical package to compute relevant probabilities
associated with its distribution.
Recall the distibution of the sample variance,
for a simple random sample from a Normal distribution;
understand that it is independent of the sample mean; and use statistical tables
or an appropriate statistical package to compute relevant probabilities
associated with its distribution.
Identify the distribution of a sum or linear combination of independent
Normally distributed random variables, and use statistical tables
or an appropriate statistical package to compute relevant probabilities
associated with its distribution.
Identify the distribution of a sum of squares of independent
random variables, each with the standard Normal distribution, and use statistical tables
or an appropriate statistical package to compute relevant probabilities
associated with its distribution.
- Identify the distribution of a sum of independent
random variables, each with the same Exponential distribution, and use statistical tables
or an appropriate statistical package to compute relevant probabilities
associated with its distribution.
Define the chi-square and t distributions,
and look up percentile points of each distribution in the recommended statistical tables.
- Apply the results above to find the mean and variance of an estimator of
a parameter or other population quantity of interest,
and hence find its bias and mean square error.
Suggested Reading
| Rice | | Chapter 2 | | Sections 2.3 | | Functions of a random variable |
| | Chapter 6 | | Sections 6.1 | | Introduction |
| | | | Sections 6.2 | | Chi-squared, t and F distributions |
| | | | Sections 6.3 | | The sample mean and the sample variance |
Handouts and Problem Sheets
Copies of Handouts, Problem Sheets and Solution Sheets for the unit
will be made available each week on the Statistics 1 course pages on
Blackboard.
Questions - set this week
PROBLEM SHEET 6 -- Questions 2, 3, 4
Interesting links
Statistical Java
The Statistical Java site is supported by the Department of Statistics at Virginia Tech,
with the aim of providing an interactive environment for teaching statistics.
From the drop down menus on the home page you can access a variety of illustrative applets
with accompanying information pages.
Particularly relevant for this week are the applet and page on the t-distribution
(select Statistical Theory | T Probabilities | Main Page or Applet or Applet Instructions).
The applet aims to allows you to see how the t-distribution is related to the standard
normal distribution by calculating relevant probabilities.
There are also applets and pages specifically related to individual distributions,
including the t and Chi-square distributions
(select Statistical Theory | Probability Distributions | Main Page and then select the relevant
distribution and applet from the explanatory text).
Finally, pages and applets relevant to last week's section on the Central Limit Theorem can be found
by selecting Statistical Theory | Central Limit Theorem
Note that I have no control over the content or availability of these external web pages.
The links may be slow to load, or may sometimes fail altogether - please email me to report if a link goes down.
Similarly applets may be slow to load or run, but beware that
you may experience problems if you try to exit them before they have finished loading.
Return to the Statistics1 information page
Dr E J Collins,
Department of Mathematics,
University of Bristol, Bristol, BS8 1TW, UK
Email: E.J.Collins@bristol.ac.uk
Telephone: +44 (0) 117 928 7977; Fax: +44 (0) 117 928 7999