Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS

the Gaussian distribution. Explicitly,

Pr(X=x)≈

1

σ

√

2 π

∫x+0. 5

x− 0. 5

exp

[

−

1

2

(u−μ

σ

) 2 ]

du.

The Gaussian approximation is particularly useful for estimating the binomial

probability thatXlies between the (integer) valuesx 1 andx 2 ,

Pr(x 1 <X≤x 2 )≈

1

σ

√

2 π

∫x 2 +0. 5

x 1 − 0. 5

exp

[

−

1

2

(u−μ

σ

) 2 ]

du.

A manufacturer makes computer chips of which10%are defective. For a random sample

of 200 chips, find the approximate probability that more than 15 are defective.

We first define the random variable

X= number of defective chips in the sample,

which has a binomial distributionX∼Bin(200, 0.1). Therefore, the mean and variance of
this distribution are

E[X] = 200× 0 .1 = 20 and V[X] = 200× 0. 1 ×(1− 0 .1) = 18,

and we may approximate the binomial distribution with a Gaussian distribution such that
X∼N(20, 18). The standard variable is

Z=

X− 20

√

18

,

and so, usingX=15.5 to allow for the continuity correction,

Pr(X> 15 .5) = Pr

(

Z>

15. 5 − 20

√

18

)

=Pr(Z>− 1 .06)

=Pr(Z< 1 .06) = 0. 86 .

Gaussian approximation to the Poisson distribution

We first met the Poisson distribution as the limit of the binomial distribution for

n→∞andp→0,takeninsuchawaythatnp=λremains finite. Further, in

the previous subsection, we considered the Gaussian distribution as the limit of

the binomial distribution whenn→∞butpremains finite, so thatnp→∞also.

It should come as no surprise, therefore, that the Gaussian distribution can also

be used to approximate the Poisson distribution when the meanλbecomes large.

The probability function for the Poisson distribution is

f(x)=e−λ

λx

x!

,

which, on taking the logarithm of both sides, gives

lnf(x)=−λ+xlnλ−lnx!. (30.115)