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
W W
W (a X bY )
2 2 2 W (a 2 X b 2 Y )
6) For a normal distriband kurtosis (K) is 3.
2. The Standard Normal Distribution
5) A linear combination (function) of two (or more) normally distributed random variables is itself normally distributed. X and Y are independent, X ~ N ( , 2 ) X X Y ~ N ( , 2) Y Y W=aX+bY, then W ~ N[ , 2 ]
· 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.
X ~ N (, 2 / n)
A standard normal variable:
Z
X

n
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.
Z
X X
Z~N(0,1)
X
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.
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.
4.2 THE SAMPLING , OR PROBABILITY,
DISTRIBUTION OF THE SAMPLE MEAN
1. The sample mean and its distribution
X
（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.
（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.