9.3 Moments of Distributions

107

23, it is lower still, approximately 0.049. This example highlights the difficulties

of drawing inferences from small samples. Two failures out of 23 is slightly better

evidence in favour of the vaccine than no failures out of 10.

9.3

Moments of Distributions

A random variable is “a function defined on a sample space” (e.g., the number

of successes in nn Bernoulli trials). A unique rule associates a number bold upper XX with any

sample point. The aggregate of all sample points on which bold upper XX assumes the fixed

value x Subscript jx j forms the event that bold upper X equals x Subscript jX = x j, with probability upper P left brace bold upper X equals x Subscript j Baseline right braceP{X = x j}. ¹¹ The function

f left parenthesis x Subscript j Baseline right parenthesis equals upper P left brace bold upper X equals x Subscript j Baseline right brace f (x j) = P{X = x j} is called the (probability) distribution of the random variable

bold upper XX. ¹² Joint distributions are defined for two or more variables defined on the same

sample space. For two variables, p left parenthesis x Subscript j Baseline comma y Subscript k Baseline right parenthesis equals upper P left brace bold upper X equals x Subscript j Baseline comma bold upper Y equals y Subscript k Baseline right bracep(x j, yk) = P{X = x j, Y = yk} is the joint prob-

ability distribution of bold upper XX and bold upper YY.

The mean, average, or expected value of bold upper XX is defined by ¹³

upper F le f

t

pa re nthesis x right parenthesis equals upper P left brace bold upper X less than or equals x right brace equals sigma summation Underscript x Subscript j Baseline less than or equals x Endscripts f left parenthesis x Subscript j Baseline right parenthesis

(9.33)

provided that the series converges absolutely. The expectation of the sum (or product)

of random variables is the sum (or product) of their expectations. Proofs are left to

the reader.

Any function of bold upper XX may be substituted for bold upper XX in definition (9.33), with the same

proviso of series convergence. The expectations of the rrth powers of bold upper XX are called

therrth moments ofbold upper XX about the origin. ¹⁴ SinceStartAbsoluteValue bold upper X EndAbsoluteValue Superscript r minus 1 Baseline less than or equals StartAbsoluteValue bold upper X EndAbsoluteValue Superscript r Baseline plus 1|X|^r−1 ≤|X|^r + 1, if therrth moment

exists, so do all the preceding ones. The expectation of the square of bold upper XX’s deviation

from its mean value has a special name, the variance: ¹⁵

sigma Subscript upper X Superscript 2 Baseline equals Var left parenthesis bold upper X right parenthesis equals bold upper E left parenthesis left parenthesis bold upper X minus bold upper E left parenthesis bold upper X right parenthesis right parenthesis squared right parenthesis equals bold upper E left parenthesis bold upper X squared right parenthesis minus bold upper E left parenthesis bold upper X right parenthesis squared periodσ²

X ^{= Var(X) = E((X −E(X))2) = E(X2) −E(X)2 .}

(9.34)

Its positive square rootsigmaσ is called the standard deviation ofbold upper XX, hinting at its use as a

rough measure of spread. The mean and variance (i.e., the first and second moments)

11bold upper XX may assume the valuesx 1 comma x 2 comma ellipsisx1, x2, . . . (i.e., the range ofbold upper XX).

12 The distribution functionupper F left parenthesis x right parenthesisF(x) ofbold upper XX is defined by

upper F left parenthesis x right parenthesis equals upper P left brace bold upper X less than or equals x right brace equals sigma summation Underscript x Subscript j Baseline less than or equals x Endscripts f left parenthesis x Subscript j Baseline right parenthesisF(x) = P{X ≤x} =

E

x j ≤x

f (x j)

(9.32)

(i.e., a nondecreasing function tending to 1 asx right arrow normal infinityx →∞).

13 Also denoted by angular brackets or a bar.

14 Notice the mechanical analogies: centre of gravity as the mean of a mass and moment of inertia

as its variance.

15 Older literature uses the term “dispersion”.