Transcript The binomial distribution

The binomial
distribution
(Session 06)
SADC Course in Statistics
Learning Objectives
At the end of this session you will be able to:
• describe the binomial probability
distribution including the underlying
assumptions
• calculate binomial probabilities for simple
situations
• apply the binomial model in appropriate
practical situations
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Study of child-headed households
• One devastating effect of the HIV and AIDS
pandemic is the emergence of child-headed
households, i.e. ones where both parents have
died and the children are left to fend for
themselves.
• Suppose it is of interest to study in greater
detail those households that are child-headed.
• Statistical techniques that may be employed
require initially, a knowledge of the
distributional pattern of the random variable X
corresponding to the number of child-headed
households.
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A probability distribution for X
• Interest is on the distribution of
X = number of child-headed households
• Under certain assumptions, X has a
binomial distribution.
• To introduce this distribution, we first
deal with a simpler (but related)
distribution
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The Bernoulli Distribution
The simplest probability distribution is one
describing the behaviour of a dichotomous
(binary) random variable, i.e. one with two
possible outcomes; (Success, Failure), (Yes,
No), (Female, Male), etc.
Probability
Success
Values of random
variable
1
failure
0
1-p
Total
1
Outcome
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p
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Background
In general, we have a sequence of n trials,
each with just two possible outcomes.
e.g. visiting n households in turn and
recording whether it is child-headed.
Call one outcome a “success”, the other a
“failure”. Let probability (of a success) = p.
The word success is a generic term used to
represent the outcome of interest, e.g. if a
sampled household is child-headed we call it
a “success” because that is the outcome of
interest.
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Basics and terminology
Let X be the number of successes in n trials.
X is said to have a binomial distribution if:
• The probability of success p is the same
for each trial.
• The trials have independent outcomes.
In the context of our example, X=number of
child-headed HHs from n HHs sampled.
p=probability of a HH being child-headed.
Under what conditions would X be binomial?
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Binomial Probability Distribution
The probability of finding k successes out
of n trials is given by
n!
P( X  k ) 
p k (1  p) nk ,
k!(n  k )!
k  0,1,, n
Here n! = n(n-1)(n-2)……. (3) (2) (1); 0!=1.
Thus, for example, 4! = 4 x 3 x 2 x 1 = 24.
Exercise: If p=0.2 and n=10, confirm that
10 !
3
10 3
P( X  3 ) 
0.2 ( 1  0.2 )  0.2
3!( 10  3 )!
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Binomial Probability Distribution
In computing binomial probabilities, the value
of p is often unknown.
It is then estimated by the proportion of
successes in the sample.
observed no.of
 successes( sayr )inntrials
i.e. p̂
totalsamplesize,n
r
i.e. p̂
n
Following graphs show binomial probabilities
for n=10 and differing values of p.
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Example 1: Left-handedness
Suppose the
probability of a
person being lefthanded is p=0.2.
Binomial distribution with p = 0.2
0.35
Probability
0.30
0.25
Let X be number
left-handed
persons in a
group of 10.
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
7
8
9
10
X
There are 11 possible outcomes.
Graph shows P(X=2)=0.3,
P(X=3)=0.2, P(X>6) almost=0.
Graph shows
probability of 0,
1, 2, … lefthanded persons
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Example 2: Tossing a coin
A coin is tossed.
The probability of
getting a head is
p=0.5.
Binomial distribution with p = 0.5
0.30
Probability
0.25
0.20
Let X be number
heads in 10
tosses of the coin.
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
7
8
9
10
X
The distribution is symmetrical.
We find P(X=2)=P(X=8)=0.044,
P(X=3)=P(X=7)=0.12, etc.
Graph shows
probability of
getting 0, 1, 2, …
heads.
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Example 3: Selecting a rural village
Ratio of rural
villages to urban
villages is 4:1.
Binomial distribution with p = 0.8
0.35
Probability
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
7
8
X
The distribution is now
concentrated to the right.
Here P(X<4) is almost zero.
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10
Suppose 10
villages are
selected at
random. Let X be
number of rural
villages selected.
Graph shows
probability of
getting 0, 1, 2, …
rural villages.
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Properties of Binomial Distribution
The mean (average) of the binomial distribution
with parameters n and p = np.
e.g. In a population of size 1000, suppose the
probability of selecting a child-headed HH is
p=0.03. Then the mean number of child-headed
HHs is 1000x0.03 = 30.
Recall that the mean = expected value of X. Thus
n
n!
  E( X )   x
p x (1  p ) n  x  np.
x!(n  x)!
x 0
Note: Since the binomial is a probability distribution,
n
n!
x
n x
p (1  p)  1.

x  0 x!( n  x )!
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Further Properties:
The standard deviation of the binomial
distribution is np(1  p)
For n=1000, p=0.2 the standard deviation is
therefore np(1  p) =[1000*0.2*0.8]½ = 12.65
The theoretical derivation is given below.
n
n!
E( X )   x
p x ( 1  p )n  x
x!( n  x )!
x 0
2
2
Above can be shown to be np( 1  p )  n 2 p 2 .
Var( X )   2  E( X 2 )   2  np( 1  p ).
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Practical work follows to
ensure learning objectives
are achieved…
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