1. Lecture 2. The Binomial Distribution
Outlines for Today
The Equi-probability Model
The Binomial Distribution
11/17/2018
SA3202, Lecture 2

2. The Equi-probability Model: the model postulating “all the categories are equally likely”.
For the Random Number Data, Question of Interest: Are the “Random Digits” generated by the calculator genuinely random?
This is equivalent to test if the digits follow an equi-probability model.
Number of possible digits:
Expected frequency for a digit:
Observed frequency for a digit:
(E-O)^2/E:
2*O*log(O/E):
df of the Pearson’s goodness of fit test statistic:
df of the Wilk’s likelihood ratio test statistic:
Table values:
Conclusions:
11/17/2018
SA3202, Lecture 2

3. The Binomial Distribution
The binomial distribution is an important distribution in categorical data analysis.
Bernoulli Trial: a trial with two different outcomes: “success”(S) and “failure”(F)
e.g.
Let p be the probability of the event “success”. Denote X be a random variable which takes value 1 when the Bernoulli trial is “success”, and 0 when “failure”. That is,
P(X=1)=P(“S”)=p,
P(X=0)=P(“F”)=1-p=q.
X is called the Bernoulli variable, denoted as X~ Binom(1,p).
11/17/2018
SA3202, Lecture 2

4. The Binomial Distribution
Binomial R.V. the sum of several Beroulli random variables X1,X2,…Xn, i.e.
S=X1+X2+…+Xn~Binom(n,p)
with index n and parameter p. A Beroulli r.v. is a special binomial r.v. with index 1 and parameter p.
Binomial Distribution the distribution of a binomial r.v. :
P(X=k)=P( k “success” in n Bernoulli trials)=
where q=1-p and is the binomial coefficients.
11/17/2018
SA3202, Lecture 2

5. The Binomial Distribution
Example 1: Assume there are about 5% of the students in NUS who are smokers. Let X be 1 if a student is smoker and 0 otherwise. Then X~Binom( ). Let Y be the number of smokers in a class which has 50 students. Then Y~Binom( ).
Example 2: Assume there are about 35% of the Singaporean who have visited US. Let X be 1 if a Singaporean who has visited US and 0 otherwise. Then X~Binom( ). Let Y be the number of Singaporean in a community which has 2000 Singaporean. Then Y~Binom( ).
The Dual Interpretation of p:
(a). The “population proportion” that possesses some property
(b). The “probability” that a member chosen at random possesses some property
11/17/2018
SA3202, Lecture 2

6. The Binomial Distribution
Expectation and Variance of X~Binom(n,p)
E(X)=np, Var(X)=npq
Interpretation of the mean E(X)=np
The Expected Frequency= # of Repeats x Probability
The Expected Frequency = Sample Size x Population Proportion
e.g. When a fair coin is tossed 100 times, the expected # of times that a “head” appears is
e.g. If the failure proportion is 5% for a course, then it is expected that there are students in the class who has 100 students have failed the course.
11/17/2018
SA3202, Lecture 2

7. The Binomial Distribution
Asymptotic Normality X~Binom(n,p)
X~AN(np,npq)
provided p is not very close to 0 or 1.
11/17/2018
SA3202, Lecture 2

8. The Binomial Distribution
Estimation of p Given X~Binom(n,p), it is natural to estimate p by the sample proportion
p=X/n.
Expectation and Variance of the sample Proportion
E( )=p, Var( )=pq/n
The Estimated Variance and the Standard Deviation (s.e.) of p^:
Var( )=p q/n, s.e.( )=(p q/n)^(1/2).
11/17/2018
SA3202, Lecture 2

9. The Binomial Distribution
Asymptotic Normality of the sample Proportion
~ AN(p, pq/n)
Confidence Interval: an interval that contains a true parameter with a given confidence level.
the estimate table value x s.e. (the estimate)
The table value is determined by the distribution of the estimate and the given confidence level.
11/17/2018
SA3202, Lecture 2

10. The Binomial Distribution
The Approximation CI for p :
Example In a sample of 200 students, 120 are in favor of a particular proposal. Then
The estimated proportion=
The estimated s.e. =
The CI of 95% is
11/17/2018
SA3202, Lecture 2