ABSTRACT
In this chapter we study the distribution of the sample mean of a random sample taken from an arbitrary distribution (cf. Definition 5.5). The exact distribution of the sample mean is, in general, not an easy matter to determine, except for some important special cases. For example, it can be shown that the sample mean taken from a normal distribution is again normally distributed with mean µ and standard deviation σ/
√ n. The central limit
theorem asserts that for large values of the sample size n, n > 30, say, the distribution of the sample mean is approximately normal with mean µ and standard deviation σ/
√ n. We
then introduce other distributions, such as the chi-square (denoted χ2), t, and F distributions related to the normal distribution as well as to others. For example, the chi-square distribution is a special case of the gamma distribution, and it can be shown that t2 has an F distribution (Section 6.3.2). Organization of Chapter
1. Section 6.2: Sampling from a Normal Distribution
2. Section 6.3: The Distribution of the Sample Variance
3. Section 6.3.1: Student’s t Distribution
4. Section 6.3.2: The F Distribution
5. Section 6.4: Mathematical Details and Derivations
The term sampling from a normal population refers to the important special case when the parent df is a normal distribution. Snedecor and Cochran (1980), Statistical Methods, 7th ed., Iowa State University Press, Ames, Iowa, list several situations in which it is reasonable
and Finance
are imately normal.
