Download Binomial Distribution

Chapter 1
Discrete Probability
Distribution: Binomial
Distribution
ІМќ
INSTITUT MATEMATIK
KEJURUTERAAN
U N I M A P
Binomial Distribution

Binomial distribution is the probability
distribution of the number of successes in n
trials.

E.g.
1.
2.
3.
Lecture 4
No. of getting a head in tossing a coin 10 times.
No. of getting a six in tossing 7 dice.
No of missile hits the target.
Abdull Halim Abdul
2
Binomial distribution is characterised by
1. the number of trial, n and
2. the probability of success in each trial, p
And is denoted by B(n,p)
Lecture 4
Abdull Halim Abdul
3

If a random variable X is distributed binomial
with the parameter n and p then

X ~ B(n,p)
The probability distribution of X is
 P(X=x) = nCxpx(1-p)n-x


Lecture 4
*Page 221 (text book)
*in text book π is used instead of p
Abdull Halim Abdul
4
Example
If X~B(4,0.1) then
P(X=3) = 4C3(0.1)3(0.9)1 = 0.0036
P(X=4) = 4C4(0.1)4(0.9)0 = 0.0001
P(X≥4) = P(X=3)+P(X=4) = 0.0037
What if n is large? Calculation would be tedious.
Solution… using cummulative binomial distribution
table or statistical software or excel spreadsheet.
Lecture 4
Abdull Halim Abdul
5

cummulative binomial distribution table
already discussed at school. It will only be
discussed in the tutorial.

Using excel will be discussed using a few
examples.
Lecture 4
Abdull Halim Abdul
6
Poisson Distribution

Poisson distribution is the probability
distribution of the number of successes in a
given space*.


*space can be dimensions, place or time or combination of
them
E.g.
1.
2.
3.
Lecture 4
No. of cars passing a toll booth in one hour.
No. defects in a square meter of fabric
No. of network error experienced in a day.
Abdull Halim Abdul
7
Poisson distribution is characterised by the
mean success, λ.
And is denoted by Po(λ)
Lecture 4
Abdull Halim Abdul
8

If a random variable X is distributed Poisson
with the parameter λ then

X ~ Po(λ)
The probability distribution of X is
 x
e


P( X  x) 
x!
*Page 228 (text book)
Lecture 4
Abdull Halim Abdul
9
Example
If X~ Po(3) then
P(X=2) =
e 3 32
2!
= 0.2240
P(X>2) = P(X=3)+P(X=4)+…+P(X=∞)
= 1 – [P(X=0)+P(X=1)+P(X=2)] = 0.5768
To avoid tedious calculation it is easier to use
cummulative Poisson distribution table or statistical
software or excel spreadsheet.
Lecture 4
Abdull Halim Abdul
10

cummulative binomial distribution table
already discussed at school. It will only be
discussed in the tutorial.

Using excel will be discussed using a few
examples.
Lecture 4
Abdull Halim Abdul
11
Why Normal Distribution



Numerous continuous variables have
distribution closely resemble the normal
distribution.
The normal distribution can be used to
approximate various discrete prob. dist.
The normal distribution provides the basis for
classical statistical inference.
Lecture 4
Abdull Halim Abdul
12
Properties of Normal
Distribution




It is symmetrical with mean, median and
mode are equal.
It is bell shaped
Its interquartile range is equal to 1.33 std
deviations.
It has an infinite range.
Lecture 4
Abdull Halim Abdul
13
Normal distribution is characterised by its
mean, μ and its std deviation, σ.
And is denoted by N(μ , σ2)
Lecture 4
Abdull Halim Abdul
14

If a random variable X is distributed normal
with the mean, μ and its std deviation, σ then

X ~ N(μ , σ2)
The probability distribution function of X is
f ( x) 
1
e
 2
1  X  
 

2  
2
*Page 250 (text book)
Lecture 4
Abdull Halim Abdul
15
If X ~ N(μ , σ2)


Then P(X = a) = 0, where a is any constant.
Previuosly it is very difficult to calculate the
probability using the pdf. So all normal
distribution is converted to std normal
distribution, Z ~ N(0,1) for calculation.
i.e
Lecture 4
Z
X 

Abdull Halim Abdul
16

Calculation using std normal distribution table
already discussed at school. It will only be
discussed in the tutorial.

Using excel will be discussed using a few
examples.
Lecture 4
Abdull Halim Abdul
17