Mathematical Methods for Physicists: A concise introduction - Site Map

INTRODUCTION TO PROBABILITY THEORY
By analogy with those for discrete random variables the expected value or mean
and the variance of a continuous random variable X with probability
density function f …x† are de®ned, respectively, to be:
E…X† ˆ ˆ
Z 1
1
xf …x†dx;
…14:40†
Var…X† ˆ 2 ˆ E……X † 2 †ˆ
Z 1
1
…x † 2 f …x†dx:
…14:41†
The Gaussian (or normal) distribution
One of the most important examples of a continuous probability distribution is
the Gaussian (or normal) distribution. The density function for this distribution is
given by
f …x† ˆ p 1
e …x†2 =2 2 ; 1< x < 1; …14:42†
2
where and are the mean and standard deviation, respectively. The corresponding
distribution function is
F…x† ˆP…X x† ˆ p 1
2
Z x
1
e …u†2 =2 2 du:
…14:43†
The standard normal distribution has mean zero … ˆ 0† and standard deviation
( ˆ 1)
f …z† ˆp 1 e z2 =2 : …14:44†
2
Any normal distribution can be `standardized' by considering the substitution
z ˆ…x †= in Eqs. (14.42) and (14.43). A graph of the density function (14.44),
known as the standard normal curve, is shown in Fig. 14.7. We have also indicated
the areas within 1, 2 and 3 standard deviations of the mean (that is between
z ˆ1 and ‡1, 2 and ‡2, 3 and‡3):
P…1 Z 1† ˆp
1
2
P…2 Z 2† ˆp
1
2
P…3 Z 3† ˆp
1
2
502
Z 1
1
Z 2
2
Z 3
3
e z2 =2 dz ˆ 0:6827;
e z2 =2 dz ˆ 0:9545;
e z2 =2 dz ˆ 0:9973: