Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

xf(x) (binomial) f(x) (Gaussian)

0 0.0001 0.0001

1 0.0016 0.0014

2 0.0106 0.0092

3 0.0425 0.0395

4 0.1115 0.1119

5 0.2007 0.2091

6 0.2508 0.2575

7 0.2150 0.2091

8 0.1209 0.1119

9 0.0403 0.0395

10 0.0060 0.0092

Table 30.4 Comparison of the binomial distribution forn=10andp=0. 6

with its Gaussian approximation.

to obtain

f(x)≈

1

√

2 πn

(x

n

)−x− 1 / 2 (n−x

n

)−n+x− 1 / 2

px(1−p)n−x

=

1

√

2 πn

exp

[

−

(

x+^12

)

ln

x

n

−

(

n−x+^12

)

ln

n−x

n

+xlnp+(n−x)ln(1−p)

]

.

By expanding the argument of the exponential in terms ofy=x−np,where

1 ynpand keeping only the dominant terms, it can be shown that

f(x)≈

1

√

2 πn

1

√

p(1−p)

exp

[

−

1

2

(x−np)^2

np(1−p)

]

,

which is of Gaussian form withμ=npandσ=

√

np(1−p).

Thus we see that thevalueof the Gaussianprobability density functionf(x)is

a good approximation to theprobabilityof obtainingxsuccesses inntrials. This

approximation is actually very good even for relatively smalln. For example, if

n=10andp=0.6 then the Gaussian approximation to the binomial distribution

is (30.105) withμ=10× 0 .6=6andσ=

√

10 × 0 .6(1− 0 .6) = 1.549. The

probability functionsf(x) for the binomial and associated Gaussian distributions

for these parameters are given in table 30.4, and it can be seen that the Gaussian

approximation is a good one.

Strictly speaking, however, since the Gaussian distribution is continuous and

the binomial distribution is discrete, we should use the integral off(x)forthe

Gaussian distribution in the calculation of approximate binomial probabilities.

More specifically, we should apply acontinuity correctionso that the discrete

integerxin the binomial distribution becomes the interval [x− 0. 5 ,x+0.5] in