Important Probability Distributions Discrete Probability Distributions Binomial Distribution A DRV X follows a binomial distribution,denoted as B(n,p), if its PMF fx(x)=()p(1-p)m-m,x=0,1,,n. where n≥1and0<p<1. Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 11/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 11/135 Binomial Distribution Discrete Probability Distributions
Important Probability Distributions Discrete Probability Distributions Binomial Distribution Remarks: .A binomial random variable can take n+1 possible in- teger values {0,1,..,n. Question:When can this distribution arise? Suppose one throws a coin n times independently.How many heads can one get from these n trials? Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 12/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 12/135 Binomial Distribution Discrete Probability Distributions Remarks:
Important Probability Distributions Discrete Probability Distributions Binomial Distribution Remarks: Let X;=the number of heads for the i-th trial,and let X the total number of heads in the n trials.Then X=>Xi i=1 where Xi is a sequence of IID Bernoulli(p)random vari- ables.It can be shown that X follows a B(n,p)distri- bution. To be Continued Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 13/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 13/135 Binomial Distribution Discrete Probability Distributions Remarks: To be Continued
Important Probability Distributions Discrete Probability Distributions Binomial Distribution Question:How to verify ∑fx(e)=∑()p'(1-p)m-r=1 for all n≥1 and all p∈(0,1)？ C=0 c=0 By the binomial theorem that for any real x and y and integer n≥0， m (x+)”=()xy”-i, 2=0 we can obtain the identity by setting x =p and y= 1-p.This is the reason why the distribution is called the binomial distribution. To be Continued Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 14/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 14/135 Binomial Distribution Discrete Probability Distributions To be Continued
Important Probability Distributions Discrete Probability Distributions Binomial Distribution E(X) =∑f() x=0 = ∑x(G)p(1-p)n-t c=0 ●Mean of B(n,p): = x()p(1-p）n-x+0.(6)m(1-p)n-0 c=1 ∑n(-p(1-p)m- x=1 m-1 np ∑(g-1)p(1-p）n-)- y=0 n-1 np ∑fr()=np. To be Continued y=0 Important Probability Distributions Introduction to Statistics and Econometrics June25,2019 15/135
Important Probability Distributions Important Probability Distributions Introduction to Statistics and Econometrics June 25, 2019 15/135 Binomial Distribution Discrete Probability Distributions To be Continued