Important Probability Distributions Discrete Probability Distributions Binomial Distribution 。Variance of B(n,p): - np ∑yg-1)p(1-p)m-10-y y=0 E(X2) =∑a2fx() m-1 x=0 +np∑g-1p(1-p)m-)-y y=0 ∑x2()p(1-p)m- n-1 x=0 npE(Y)+np∑fr(g) ∑x·x)p产(1-p)m- y=0 np(n-1)p+np x=1 np[(m-1)p+1: ∑emg=p'(1-pjm-4 x= m-1 n(y+1)(g-1)p+1(1-p)m-1)-y 2=0 To be Continued Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 16/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 16/135 Binomial Distribution Discrete Probability Distributions To be Continued
Important Probability Distributions Discrete Probability Distributions Binomial Distribution 。Variance of B(n,p): It follows that oR=E(X2)-候 {np·[(n-1)p]+np}-(np)2 np(1-p). To be Continued Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 17/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 17/135 Binomial Distribution Discrete Probability Distributions To be Continued
Important Probability Distributions Discrete Probability Distributions Binomial Distribution MGF of B(n,p): Mx(t)= E(etx) m 入er()p(1-p)n-w x=0 》)(pe)(1-p)m-4 (pe+1-p)r. Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 18/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 18/135 Binomial Distribution Discrete Probability Distributions
Important Probability Distributions Discrete Probability Distributions Binomial Distribution B(n,p)has been one of the oldest to have been the sub- ject of study in probability and statistics. It has wide applications in economics: -In quality control,for example,it can be used to approximate the distribution of the numbers of de- fective products in a total of n products each of which is probability p to be defective. It can also be used to model the cumulative num- ber of jumps occurred in financial price movements during a given period of time (e.g.,Das 2002). Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 19/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 19/135 Binomial Distribution Discrete Probability Distributions
Important Probability Distributions Discrete Probability Distributions Negative Binomial Distribution B(n,p)describes the probability distribution for the num- ber of successes in a fixed number of n trials. Negative Binomial,NB(n,p),characterizes the proba- bility distribution of the number of trials required to obtain a given number of successes. In a sequence of independent Bernoulli(p)trials,let X the number of trials such that at the X-th trial the r-th success occurs,where r is a fixed integer.This implies that X-1 is the number of trials right before the r-th success is obtained. To be Continued Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 20/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 20/135 Negative Binomial Distribution Discrete Probability Distributions To be Continued