SADC Course in Statistics
The binomial distribution
(Session 06)
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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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• 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
A probability distribution for X
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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.
Outcome Values of random variable
Probability
Success 1 p
failure 0 1-p
Total 1
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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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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 ofchild-headed HHs from n HHs sampled.
p=probability of a HH being child-headed.
Under what conditions would X be binomial?
Basics and terminology
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Binomial Probability Distribution
The probability of finding k successes out of n trials is given by
nkppknk
nkXP knk ,,1,0,)1(
)!(!
!)(
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
3 10 3103 0 2 1 0 2 0 2
3 10 3
!P( X ) . ( . ) .
!( )!
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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.
i.e.
i.e.
Following graphs show binomial probabilities for n=10 and differing values of p.
observed no.of successes ( say r ) in n trialsp̂
total sample size,n
r
p̂n
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There are 11 possible outcomes. Graph shows P(X=2)=0.3, P(X=3)=0.2, P(X>6) almost=0.
Binomial distribution with p = 0.2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 1 2 3 4 5 6 7 8 9 10
X
Pro
ba
bil
ity
Example 1: Left-handedness
Suppose the probability of a person being left-handed is p=0.2.
Let X be number left-handed persons in a group of 10.
Graph shows probability of 0, 1, 2, … left-handed persons
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The distribution is symmetrical. We find P(X=2)=P(X=8)=0.044, P(X=3)=P(X=7)=0.12, etc.
Example 2: Tossing a coin A coin is tossed. The probability of getting a head is p=0.5.
Let X be number heads in 10 tosses of the coin.
Graph shows probability of getting 0, 1, 2, … heads.
Binomial distribution with p = 0.5
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 1 2 3 4 5 6 7 8 9 10
X
Pro
ba
bil
ity
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The distribution is now concentrated to the right.
Here P(X<4) is almost zero.
Example 3: Selecting a rural village
Ratio of rural villages to urban villages is 4:1.
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.
Binomial distribution with p = 0.8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 1 2 3 4 5 6 7 8 9 10
X
Pro
ba
bil
ity
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Properties of Binomial DistributionThe 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
.)1()!(!
!)(
0
npppxnx
nxXE xnx
n
x
.1)1()!(!
!
0
xnxn
x
ppxnx
nNote: Since the binomial is a probability distribution,
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The standard deviation of the binomial distribution is
For n=1000, p=0.2 the standard deviation is therefore =[1000*0.2*0.8]½ = 12.65
The theoretical derivation is given below.
)1( pnp
2 2
0
2 2
2 2 2
1
1
1
nx n x
x
n!E( X ) x p ( p )
x!( n x )!
np( p ) n p .
Var( X ) E( X ) np( p ).
Above can be shown to be
Further Properties:
)1( pnp
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Practical work follows to ensure learning objectives
are achieved…
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