Chapter 3 SOME IMPORTANT PROBABILITY DISTRIBUTIONS
Chapter 3 SOME IMPORTANT PROBABILITY DISTRIBUTIONS
3. The Normal Distribution X-N (u, 02) x The Normal distribution the distribution of a continuous rv whose value depends on a number of factors, yet no single factor dominates the other 1. Properties of the normal distribution 1)The normal distribution curve is symmetrical around its mean value. 2) The PdF of the distribution is the highest at its mean value but tails off at its extremities 3)μ土σ68% u±2o95% μ士30997% 4)A normal distribution is fully described by its two parameters: A and g2
3.1 The Normal Distribution X～N（μ，σ 2） The Normal Distribution: the distribution of a continuous r.v. whose value depends on a number of factors, yet no single factor dominates the other. 1. Properties of the normal distribution: 1) The normal distribution curve is symmetrical around its mean valueμ. 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: μ and σ2
5)Alinear combination(function)of two(or more)normally distributed random variables is itself normally distributed X and Y are independent, X-N(uvO Y~N(y,°P W=aX+bY then WN (aux +buy) 2 x +boy 6)For a normal distribution, skewness(S)is zero and kurtosis ( K)is 3 x 2. The Standard Normal Distribution Z-N(O, 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
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 W N W W ~ ( , ) ~ ( , ) 2 2 Y Y X X Y N X N ( ) W = a X + bY ( ) 2 2 2 2 2 W = a X + b Y X X X Z − = 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
3.2 THE SAMPLING, OR PROBABILITY DISTRIBUTION OF THE SAMPLE MEAN X xk I. 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. Random sample and random variables: X1, X2,,Xn are called a random sample of size n if all these x are drawn independently from the same probability distribution(i. e, each, X, 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 Random sampling: a sample of iid random variables, a iid sample
3.2 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. Random 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. Random sampling: a sample of iid random variables, a iid sample. X
(2) Sampling, or prob, distribution of an estimator IfX, x2,.,Xn is a random sample from a normal distribution with meanuand varianceo2, then the sample mean, also follows a normal distribution with the same meanubut with a variance /n x~N=(,a2/m) A standard normal varia ble
（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 X N = n n X Z − =