Aims

Many estimators are based on the sum - or the mean value - of the observations in a random sample from an underlying population distribution. The exact distribution of these quantities may be difficulty to compute, and will usually vary with the underlying population distribution. The Central Limit Theorem gives a simple way of approximating the distribution of the sum or mean, that depends only on the population mean and the population variance. It also provides a plausible explanation for the fact that the distribution of many random variables studied in physical experiments are approximately Normal, in that their value may represent the overall addition of a number of individual randon factors.

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:

• Generate random samples from a given standard distribution using the random number generator for that distribution in a statistical package such as R.

• Understand how the performance of an estimator can be related to systematic and random error through the bias and variance of the estimator.

• Evaluate the performance of an estimator for a single quantity of interest, both qualitatively from a boxplot or histogram of estimates from repeated samples and quantitatively or numerically from summary statistics derived from the repeated samples.

• Recall the statement and the implications of the Central Limit Theorem.

• Apply the Central Limit Theorem to find the approximate distribution of the sum or mean of a random sample from a population distribution with known mean and variance.

• Apply a continuity correction to improve the approximation given by the Central Limit Theorem when the underlying variable is an integer valued random variable representing counts.

Suggested Reading

Handouts and Problem Sheets

Questions - set this week

PROBLEM SHEET 5 -- Questions 1, 3, 4

Interesting links