SOME IMPORTANT PROBABILITY DISTRIBUTIONS Chapter 4 SOME IMPORTANT PROBABILITY DISTRIBUTIONS

4.1 The Normal Distribution X～N（μ，σ2） The Normal distribution: a continuous r.v.whose value depends on a number of factors, yet no single factor dominates the others. 1. Properties of the normal distribution 1) The normal distribution curve is symmetrical around its mean value μ. 2) The PDF of the distribution is the highest at its mean value but tails off at its extremities. 3) μ±σ 68% μ±2σ 95% μ±3σ 99.7% 4) A normal distribution is fully described by its two parameters: μandσ2

5) A linear combination (function) of two (or more) normally distributed random variables is itself normally distributed. X and Y are independent, W=aX+bY, then 6) For a normal distribution, skewness (S) is zero and kurtosis (K) is 3. ) , ( ~ 2 Y X N s m 2. The Standard Normal Distribution Z~N(0,1) Note: Any normally distributed r.v.with a given mean and variance can be converted to a standard normal variable, then you can know its probability from the standard normal table.

4.2 THE SAMPLING , OR PROBABILITY, DISTRIBUTION OF THE SAMPLE MEAN 1. The sample mean and its distribution （1）The sample mean The sample mean can be treated as an r.v., and it has its own PDF. ①Random sample and random variables: ——X1, X2,..., Xn are called a random sample of size n if all these Xs are drawn independently from the same probability distribution (i.e., each, Xi has the same PDF). The Xs are independently and identically distributed, random variables，i.e. i.i.d. random variables. ·each X included in the sample must have the same PDF; ·each X included in the sample is drawn independently of the others. ②Random sampling: a sample of iid random variables, a iid sample.

（2）Sampling, or prob., distribution of an estimator If X1, X2,..., Xn is a random sample from a normal distribution with meanμand varianceσ2, then the sample mean, also follows a normal distribution with the same meanμbut with a varianceσ2/n. A standard normal variable:

2. The Central Limit Theorem The central limit theorem (CLT)—if X1,X2, ..., Xn is a random sample from any population (i.e., probability distribution) with mean μ and variance σ2 , the sample mean tends to be normally distributed with mean μ and varianceσ2/n as the sample size increases indefinitely (technically, infinitely.) The sample mean of a sample drawn from a normal population follows the normal distribution regardless of the sample size. Uniform distribution: the PDF of a continuous r.v. X on the interval from a to b. 1) The PDF 2) Mean and variance

4.3 THE CHI-SQUARE( ) ISTRIBUTION 1. Definition Chi-square probability distribution: The sum of the k squared standard normal variables (Zs) follows a chi-square probability distribution with k degrees of freedom (d.f.) Degrees of freedom (d.f.): the number of independent observations in a sum of squares. 2. Properties of the Chi-square Distribution (1) Takes only positive values; (2) A skewed distribution, the fewer the d.f. the distribution will be more skewed to the right, as the d.f. increase, the distribution becomes more symmetrical and approaches the normal distribution. (3)The expected or mean value of a chi-square is k and its variance is 2k. k is the d.f. (4)Z1and Z1 are two independent chi-square variables with k1 and k2 d.f., then their sum Z1 +Z1 is also a chi-square variable with d.f.= k1 + k2

3.The application of distribution If S2 is the sample variance obtained from a random sample of n observations from a normal population with the variance of σ2, then :

The t distribution is symmetric; 1. Definition If we draw random samples from a normal population with mean μ and variance σ2 but replaceσ2 by its estimator S2, the sample mean follows the t distribution. 2. Properties of the t distribution The t distribution is symmetric; The mean of the t distribution is zero but its variance is k/(k-2). The t distribution is flatter than the normal distribution. The t distribution, like the chi-square distribution, approaches the standard normal distribution as the d.f. increase.

4.5 THE F DISTRIBUTION Definition Let X1,X2, ..., Xn be a random sample of size m from a normal population with mean and variance , and let Y1,Y2, ..., Yn be a random sample of size n from a normal population with mean and variance . Assume that the two samples are independent, suppose we want to find out if the variances of the two normal populations are the same, that is, whether . Suppose we just obtain the estimators of the variances: Then we can make a conclusion when we get the F value: If F=1, then the two population variances are the same; If F≠1,the two population variances are different. （In computing the F value, we always put the larger of the two variances in the numerator.）

2. Properties of the F distribution · The F distribution is skewed to the right and ranges between 0 and infinity; · The F distribution approaches the normal distribution as k , the d.f. become large. ·

4.6 RELATIONSHIPS AMONG THE t, F, AND AND THE NORMAL DISTRIBTUIONS 1. For large degrees of freedom (at least 30), the t distribution can be approximated by the standard normal distribution. 2. For large denominator d.f., the numerator d.f.(=m) times the F value is approximately equal to the chi-square value with the numerator d.f.(=m). 3. If Z~N(0,1) and are independent random variables from the standard normal distribution and the chi-square distribution with m d.f., respectively, then That is, the ratio of a standardized normal variable to the square-root of a chi-square variable divided by its d.f.(m) follows the t distribution with m d.f., provided the two random variables are independently distributed. 4. 5. For sufficiently large d.f., the chi-square distribution can be approximated by the standard normal distribution as follows: where k denotes the d.f.