Chapter 5: The Basic Concepts of Statistics

5.1 Population and Sample Definition 5.1 A population consists of the totality of the observations with which we are concerned. Definition 5.2 A sample is a set of the observations that constitute part or all of a population.

Definition 5.3 A random sample is a sample in which any one indivi- dual measurement in the population is as likely to be included as any other. Definition 5.4 Any function of the random variables constituting a random sample is called a statistic.

Example 5.1 are not statistics. Suppose is a sample taken from which depends a normal distribution then on unknown parameters and is a statistic, butand

5.2 Empirical distribution function

Example 5.2 Given a set of sample values : 3.2,2.5,-4,2.5,0,3,2,2.5,4,2. Then

5.3 Some Important Statistics 1. sample mean

2. sample variance 3. sample standard deviation S.

4. k-th sample moment 5. k-th sample central moment

5.4 Sampling Distributions Definition 5.5 The probability distribution of a statistic is called a sampling distribution.

5.4.1 Chi-squared Distribution Let, Theorem 5.1 the distribution of the random variable, where is given by the density function This is known as the chi-squared distribution with degrees of freedom.Denoted by

Properties:, respectively, then If X and Y are independent and

Theorem 5.2 If are independent and respectively, then for any constants.

Corllary 5.1 then (2) be a sample from Let (1)

Given, the upper-point of a PDF F is determined from the equation. Definition 5.6

Notes: represent the t-value above1. which we find an area equal to leaves an area of to the right. If then 2.

Assume be a sample from find Example 5.3 Assume two independent samples from. Example 5.4 are given,find where are the sample means.and

t-Distribution Theorem 5.3 Let X ~N(0,1) ， Y~. If X and Y are independent, then the distribution of the random variable T,where is given by the density function freedom. denoted by This is known as the t-distribution with degrees of

Notes: 1. represent the t-value above which we find an area equal to leaves an area of to the right

F-Distribution Theorem 5.4 Then the distribution of the random variable is given by the density degrees of freedom. Denoted by This is known as the F-distribution withand

Note: represent the t-value above which we find an area equal to leaves an area of to the right.

Theorem 5.5 sample mean and sample variance then (2) (3) and are independent. Let be a sample from with (1)

Assume a sample of size n=21 from is given, find. Example 5.5

Corollary 5.2 sample mean and sample variancethen Let be a sample from with

random samples, respectively, from and the distributions, Then. Corllary 5.3 Let and be independent

random samples, respectively, from and distributions,Then. Corllary 5.4 Let and be independent.

Let be a sample from Find the distribution of. Example 5.6