Fundamentals of Probability and Statistics for Engineers

alternative that the probability distribution of X is not of the stated type on the
basis of a sample of size n from population X. One of the most popular and most
versatile tests devised for this purpose is the chi-squared (^2 )goodness-of-fit
test introduced by Pearson (1900).

10.2.1 The Case of K nown Parameters

Let us first assume that the hypothesized distribution is completely specified
with no unknown parameters. In order to test hypothesis H, some statistic
h(X 1 ,X 2 ,...,Xn) of the sample is required that gives a measure of deviation of
the observed distribution as constructed from the sample from the hypothe-
sized distribution.
In the^2 test, the statistic used is related to, roughly speaking, the difference
between the frequency diagram constructed from the sample and a correspond-
ing diagram constructed from the hypothesized distribution. Let the range
space of X be divided into k mutually exclusive intervals A 1 ,A 2 ,..., and Ak,
and let Ni be the number of Xj falling into Ai, i 1, 2,... , k. Then, the observed
probabilities P(Ai) are given by

The theoretical probabilities P(Ai) can be obtained from the hypothesized
population distribution. Let us denote these by

A logical choice of a statistic giving a measure of deviation is

which is a natural least-square type deviation measure. Pearson (1900) showed
that, if we take coefficient c n/pi, the statistic defined by Expression (10.3)
has particularly simple properties. Hence, we choose as our deviation measure

M odel Verification 317

ˆ

observedP…Ai†ˆ

Ni

n

; iˆ 1 ; 2 ;...;k: … 10 : 1 †

theoreticalP…Ai†ˆpi; iˆ 1 ; 2 ;...;k: … 10 : 2 †

Xk

iˆ 1

ci

Ni

n

pi

 2

; … 10 : 3 †

ˆ

Dˆ

Xk

iˆ 1

n

pi

Ni

n

pi

 2

ˆ

Xk

iˆ 1

…Ninpi†^2

npi

ˆ

Xk

iˆ 1

N^2 i

npi

n:

… 10 : 4 †

i