Download THE BINOMIAL DISTRIBUTION

THE BINOMIAL DISTRIBUTION
THE BINOMIAL DISTRIBUTION
• A discrete distribution
THE NORMAL DISTRIBUTION
• A continuous distribution
APPROXIMATION OF THE BINOMIAL
BY THE NORMAL DISTRIBUTION
Page 1 of 12
Part 3
THE BINOMIAL DISTRIBUTION
CONSIDER AN EVENT A WITH A KNOWN
PROBABILITY p
EXAMPLE
• in a clinical trial the probability of
adverse events is 25%
ONE CONSIDERS n EXPERIMENTS (n patients)
EACH TIME EITHER A OCCURS OR IT DOES
NOT OCCUR
CALL i THE NUMBER OF OCCURRENCES
EXAMPLE
• Suppose 20 patients were included
• Calculate the probability that the event
occurs i times
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Part 3
THE BINOMIAL DISTRIBUTION
Probability of i side effects
(0.25)i . (0.75)20-i
P(i) =
where
=
20 x 19 x 18 x 17 x ............ x 2 x 1
i.(i-1).(i-2) .1. (20-i).(20-i-1). 2.1
(0.25)i = 0.25x0.25x.....x0.25 [i factors]
(0.75)20-i = 0.75x0.75x.....x0.75 [20-i factors]
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Part 3
THE BINOMIAL DISTRIBUTION
EXAMPLE
Probability that exactly 6 patients suffered from adverse
events
(0.25)6 . (0.75)14
P(6) =
=
20! . = 38760
6! x 14!
(0.25)6 = 0.000244
(0.75)14 = 0.017818
P(6) = 0.1686
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Part 3
THE BINOMIAL DISTRIBUTION
Other Probabilities
P(0) = 0.169
P(1) = 0.169
P(2) = 0.169
P(3) = 0.169
P(4) = 0.169
P(5) = 0.169
P(6) = 0.169
P(7) = 0.169
P(8) = 0.169
P(9) = 0.169
P(10) = 0.169
P(11) = 0.169
P(12) = 0.169
These probabilities depend upon n and p
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Part 3
THE BINOMIAL DISTRIBUTION
PARAMETERS
mean value
= n.p
variance
= n.p.(1-p)
mean value
= n.p.(1-p)
Page 6 of 12
Part 3
APPROXIMATION OF A
BINOMIAL BY A NORMAL
DISTRIBUTION
f(x) =
where
x = random variable (pulse rate)
e = 2,72
µ = mean value
σ = standard deviation
obtained by:
decreasing the class interval
increasing the sample size
Page 7 of 12
Part 3
APPROXIMATION OF A
BINOMIAL BY A NORMAL
DISTRIBUTION
The normal distribution is determined by
the mean value:
µ
the standard deviation:
σ
Any normal distribution can be simplified into a
particular normal distribution by the transformation
X-µ
Z = --------------σ
The new normal distribution has
mean value:
0
standard deviation:
1
Page 8 of 12
Part 3
APPROXIMATION OF A
BINOMIAL BY A NORMAL
DISTRIBUTION
This transformation is called a
Z-transformation
The new distribution is a
standardized normal distribution
It is denoted as
N(0,1)
Page 9 of 12
Part 3
APPROXIMATION OF A
BINOMIAL BY A NORMAL
DISTRIBUTION
PROPERTIES OF THE N(0,1) DISTRIBUTON
P[-0.67 ≤ Z ≤ 0.67]
P[-1
≤Z≤1
= 50.0 %
]
= 68.3 %
P[-1.96 ≤ Z ≤ 1.96]
= 95.0 %
P[-2
≤Z≤2
]
= 95.4 %
P[-3
≤Z≤3
]
= 99.7 %
Page 10 of 12
Part 3
APPROXIMATION OF A
BINOMIAL BY A NORMAL
DISTRIBUTION
APPLICATION OF THE N(0,1) DISTRIBUTON
Laboratory results from different laboratories (with
different normal ranges) can be made comparable if
•
the variable has a normal distribution
•
the normal ranges correspond to 95% tolerance ranges
EXAMPLE
LAB 1
LAB 2
40-80
40-76
mean
60
58
st. dev.
10
9
test result
50
50
normal range
standardized
(50-60)/10=-1
Page 11 of 12
(50-58)/9=-0.89
Part 3
APPROXIMATION OF A
BINOMIAL BY A NORMAL
DISTRIBUTION
For LARGE n the shape of a binomial distribution is
approximately the same as the shape of a normal
distribution
CONDITIONS
n.p≥ 5
n . (1-p) ≥ 5
Page 12 of 12
Part 3