(I): The Binomial Probability Distribution:
7.1. The Binomial Probability Distribution
Example:
[pic] representing the number of heads as flipping a fair coin twice.
[pic].
□ □
[pic] T T [pic] (1 combination)
□ □
[pic] H T
T H [pic] (2 combinations)
□ □
[pic] H H [pic] (1 combination)
[pic]
, [pic]
[pic] representing the number of heads as flipping a fair coin 3 times.
□ □ □
[pic] T T T [pic] (1 combination)
□ □ □
H T T
[pic] T H T [pic] (3 combinations)
T T H
□ □ □
H H T
[pic] H T H [pic] (3 combinations)
T H H
□ □ □
[pic] H H H [pic] (1 combination)
[pic]
, [pic]
[pic] representing the number of heads as flipping a fair coin n times.
Then,
n
□ □ …………… □
[pic] T T…………….T [pic]
(1 combination)
n
□ □ …………… □
H T……..……..T
[pic] n T H……… T [pic]
[pic] [pic] [pic] [pic] (n combinations)
T T …………..H
[pic]
[pic]
[pic]
Note: the number of combinations is equivalent to the number of ways as drawing i balls (heads) from n balls (n flips).
Example:
[pic] representing the number of successes over 3 trials.
[pic]
Suppose the probability of the success is [pic] while the probability of failure is [pic].
Then,
□ □ □
[pic] F F F [pic]
(1 combination)
□ □ □
S F F
[pic] F S F [pic]
(3 combinations)
F F S
□ □ □
S S F
[pic] S F S [pic]
(3 combinations)
F S S
□ □ □
[pic] S S S [pic]
(1 combination)
[pic]
, [pic]
[pic] representing the number of successes over n trials.
Then,
n
□ □ …………… □
[pic] F F…………….F [pic]
(1 combination)
n
□ □ ……… □
S F…….. .F
[pic] n F S……… F [pic]
[pic] [pic] [pic] [pic] (n combinations)
F F … .S
[pic]
[pic]
[pic]
From the above example, we readily describe the binomial experiment.
Properties of Binomial Experiment
• X: representing the number of successes over n independent identical trials.
• The probability of a success in a trial is p while the probability of a failure is (1-p).
Binomail Probability Distribution:
Let X be the random variable representing the number of successes of a Binomial experiment. Then, the probability distribution function for X is
[pic].
Properties of Binomial Probability Distribution:
A random variable X has the binomial probability distribution [pic] with parameter [pic], then
[pic]
and
[pic].
[Derivation:]
[pic]
The derivation of [pic] is left as exercise.
How to obtain the binomail probability distribution:
a) Using table of Binomail distribution.
b) Using computer
• by some software, for example, Excel or Minitab.
Online Exercise:
Exercise 7.1.1
Exercise 7.1.2
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- stat 515 chapter 4 discrete random variables
- introduction to the binomial distribution
- ban232 v
- binomial geometric distribution problems
- i the binomial probability distribution
- s1 sample exam questions biniomial distribution
- binomial probability worksheet ii
- example code for finding binomial probabilities in r
- ap stats chapter 8 notes the binomial and geometric
Related searches
- binomial probability distribution function calculator
- binomial probability calculator ti 84
- binomial probability between two numbers
- binomial probability formula step by step
- binomial probability formula
- binomial probability distribution table
- binomial probability examples
- binomial probability mass function calculator
- binomial probability density function calculator
- binomial probability distribution table pdf
- variance of the binomial distribution calculator
- explain the binomial probability formula