Fundamentals of Probability and Statistics for Engineers

all applications, we shall assume that sample X 1 ,X 2 ,...,Xn possesses the
following properties:

Property 1: X 1 ,X 2 ,...,Xn are independent.

Property 2: for all x, j 1, 2,... , n.

The random variables X 1 ,...,Xn satisfying these conditions are called a random
sample of size n. The word ‘random’in this definition is usually omitted for the
sake of brevity. If X is a random variable of the discrete type with probability
mass function (pmf) pX (x), then for each j.
A specific set of observed values (x 1 ,x 2 ,...,xn)isasetofsamplevalues
assumed by the sample. The problem of parameter estimation is one class in
the broader topic of statistical inference in which our object is to make infer-
ences about various aspects of the underlying population distribution on the
basis of observed sample values. For the purpose of clarification, the interre-
lationships among X,(X 1 ,X 2 ,...,Xn), and (x 1 ,x 2 ,...,xn) are schematically
shown in F igure 9.1.
Let us note that the properties of a sample as given above imply that certain
conditions are imposed on the manner in which observed data are obtained.
Each datum point must be observed from the population independently and
under identical conditions. In sampling a population of percentage yield, as
discussed in Chapter 8, for example, one would avoid taking adjacent batches if
correlation between them is to be expected.
A statistic is any function of a given sample X 1 ,X 2 ,...,Xn that does not
depend on the unknown parameter. The function h(X 1 ,X 2 ,...,Xn)inEquation
(9.2) is thus a statistic for which the value can be determined once the sample
values have been observed. It is important to note that a statistic, being a function
of random variables, is a random variable. When used to estimate a distribution
parameter, its statistical properties, such as mean, variance, and distribution, give
information concerning the quality of this particular estimation procedure. Cer-
tain statistics play an important role in statistical estimation theory; these include
sample mean, sample variance, order statistics, and other sample moments. Some
properties of these important statistics are discussed below.

X

X 1 X 2 Xn

x 1 x 2 xn

(sample)

(population)

(sample values)

Figure 9.1 Population, sample, and sample values

260 Fundamentals of Probability and Statistics for Engineers

.

. fXjx)ˆfXx) ˆ

pXjx)ˆpXx)