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Binomial (n=100, p=1/2) Distribution
0.10
Probability
0.08
0.06
0.04
0.02
0.00
30
40
50
Number of Successes
60
70
Normal Density Curve
0.5
 1  x   2 
1

 x  
exp  
2

2 
 2 

0.4
0.3
0.2
0.1
0.0









0.5
Probability
0.4
0.3
0.2
68 %
0.1
0.0
-4
-3
-2
-1
0
1
2
3
4
0.5
0.5
0.4
0.4
0.3
0.3
0.2
Probability
Probability
z
95 %
0.1
0.2
99.7 %
0.1
0.0
0.0
-4
-3
-2
-1
0
z
1
2
3
4
-4
-3
-2
-1
0
z
1
2
3
4
1.0
Standard Normal Density
Cumulative Distribution Function
0.8
z
Probability
  z      y  dy

0.6
0.4
 z2 
1
 z  
exp   
2
 2
0.2
0.0
-4
-3
-2
-1
0
z
1
2
3
4
Binomial (n=100, p=1/2) Distribution
0.10
Probability
0.08
0.06
0.04
0.02
0.00
30
40
50
Number of Successes
60
70
Normal Approximation to the
Binomial Distribution
• For n independent trials with success
probability p
1
1




b



a







2
2
  

Pa to b successes   














where  = np is the mean and  = (npq)1/2 is
the standard deviation.
Binomial (n=100, p=1/2) Distribution
With Normal Approximation Curve
0.10
Probability
0.08
0.06
0.04
0.02
0.00
30
40
50
Number of Successes
60
70
Binomial (n=100, p=1/2) Distribution
With Normal Approximation Curve
0.10
Probability
0.08
0.06
0.04
0.02
0.00
30
40
50
Number of Successes
60
70
Law of Large Numbers
• Informal: If n is large, the proportion of
successes in n Bernoulli trials will be very
close to p.
• Formal: For Bernoulli trials with n and p,
as n  ,
k

P  p     1
n

for all  > 0, where k is the number of
successes in the n trials.
Normal Approximation
• What happens to the binomial distribution
when p is very small or large (i.e., close to 0
or 1)?
Binomial (n=500, p=0.001) Distribution
Probability
0.6
0.4
0.2
0.0
-10
-5
0
Number of Successes
5
10
Binomial (n=500, p=0.001) Distribution
With Normal Approximation Curve
Probability
0.6
0.4
0.2
0.0
-10
-5
0
Number of Successes
5
10
Poisson Approximation to the
Binomial Distribution
• If n is large and p is small, the distribution of
the number of successes, k, in n Bernoulli trials
is largely determined by the mean  = np:
Pk   e


k
k!
• In general, the Poisson approximation to the
binomial will be excellent when n  100 and
 10.
Exercise
• 3000 people are watching a parade on a hot
summer day. Let’s assume the probability that
any one of the 3000 persons watching the
parade will collapse from heat exhaustion is
0.005, and that people collapse independent of
one another. What’s the probability that 4
people collapse?