Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

It is usual only to tabulate Φ(z)forz>0, since it can be seen easily, from

figure 30.14 and the symmetry of the Gaussian distribution, that Φ(−z)=1−Φ(z);

see table 30.3. Using such a table it is then straightforward to evaluate the

probability thatZlies in a given range ofz-values. For example, foraandb

constant,

Pr(Z<a)=Φ(a),

Pr(Z>a)=1−Φ(a),

Pr(a<Z≤b)=Φ(b)−Φ(a).

Remembering thatZ=(X−μ)/σand comparing (30.107) and (30.108), we see

that

F(x)=Φ

(x−μ

σ

)

,

and so we may also calculate the probability that the original random variable

Xlies in a givenx-range. For example,

Pr(a<X≤b)=

1

σ

√

2 π

∫b

a

exp

[

−

1

2

(u−μ

σ

) 2 ]

du (30.109)

=F(b)−F(a) (30.110)

=Φ

(

b−μ

σ

)

−Φ

(a−μ

σ

)

. (30.111)

IfXis described by a Gaussian distribution of meanμand varianceσ^2 ,calculatethe

probabilities thatXlies within 1 σ, 2 σand 3 σof the mean.

From (30.111)

Pr(μ−nσ < X≤μ+nσ)=Φ(n)−Φ(−n)=Φ(n)−[1−Φ(n)],

and so from table 30.3

Pr(μ−σ<X≤μ+σ)=2Φ(1)−1=0. 6826 ≈ 68 .3%,

Pr(μ− 2 σ<X≤μ+2σ)=2Φ(2)−1=0. 9544 ≈ 95 .4%,

Pr(μ− 3 σ<X≤μ+3σ)=2Φ(3)−1=0. 9974 ≈ 99 .7%.

Thus we expectXto be distributed in such a way that about two thirds of the values will
lie betweenμ−σandμ+σ, 95% will lie within 2σof the mean and 99.7% will lie within
3 σof the mean. These limits are called the one-, two- and three-sigma limits respectively;
it is particularly important to note that they are independent of the actual values of the
mean and variance.

There are many other ways in which the Gaussian distribution may be used.

We now illustrate some of the uses in more complicated examples.