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 − =
2. The Central limit Theorem The central limit theorem (CLT)ifX,X2,., Xn is a random sample from any population(i.e, probability distribution)with mean u and variance o2. the sample mean tends to be normally distributed with mean u and variances/ 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 rv. x on the interval from a to b 1)The PDF f=、1 <X<b 0 otherwise 2)Mean and variance b ECX) var(X) (b-a) 12
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 0 otherwise 1 ( ) = − = a X b b a f X 12 ( ) var( ) 2 E(X) 2 b a X a b − = + =
3.3 THE CHI-SQUARE(, ISTRIBUTION 兴1. Definition Chi-square x probability distribution: The sum of the k squared standard normal variables(Zs) follows a chi-square x probability distribution with k degree of freedom(d f) Degrees of freedom(d f): the number of independent observations in a sum of squares. *K 2. Properties of the chi-square Distribution Takes only positive values; A skewed distribution, the fewer the df the distribution will be more skewed to the right, as the d f increase, the distribution becomes more symmetrical and approaches the normal distribution The expected or mean value of a chi-square is k and its variance is 2k. k is the d f Zand Z are two independent chi-square variables with k, and k,df then their sum Z +Z is also a chi-square variable with df=kI+ k2 3.The application of x
3.3 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 degree 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 · Takes only positive values; · 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. · The expected or mean value of a chi-square is k and its variance is 2k. k is the d.f. · 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 2 2 2 2
3.4/ THE t DISTRIBUTION 1. Definition If we draw random samples from a normal population with mean u and variance o2 but replaced by its estimator $2, the sample mean follows the t distribution. X-4 S/√n 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
3.4 THE t DISTRIBUTION 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. S n X t / − =
3.5 THE FDIATRIBUTION Definition Let X,X,, ...,Xn be a random sample of size m from a normal population with mean ux and variance ox, and let Y,Y2, random sample of size n from a normal population with meap '> Yn be a and varlance oy 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 o, =o?. Suppose we just obtain the estimators of the varlances s=∑(x-x)s=∑ Then we can make a conclusion when we get the F value Y ∑(X1-X)2(m-1) ∑(Y1-Y)2/(n-1) If F=l, then the two population variances are the same; If F#l, the two population variances are different. (In computing the F value, we always put the larger of the two variances in the numerator
3.5 THE F DISTRIBUTION 1. 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 2 X = Y X 2 X 2 Y Y − − = 1 ( ) 2 2 m X X S i X − − = 1 ( ) 2 2 n Y Y S i Y ( ) /( 1) ( ) /( 1) 2 2 2 2 − − − − = = Y Y n X X m S S F i i Y X
x 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 1,k n)=(xn)/masn→>0
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. · · k F k t 1, 2 = F m n = ( m )/ m as n → 2 ( , )