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Erschienen in:
Elements of Probability Theory Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

3. Expectations and Moments of Random Variables

verfasst von : Ron C. Mittelhammer

Erschienen in: Mathematical Statistics for Economics and Business

Verlag: Springer New York

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Abstract

The definition of the expectation of a random variable can be motivated both by the concept of a weighted average and through the use of the physics concept of center of gravity, or the balancing point of a distribution of weights. We first examine the case of a discrete random variable and look at a problem involving the balancing-point concept.

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Vorheriges Kapitel Random Variables, Densities, and Cumulative Distribution Functions
Nächstes Kapitel Parametric Families of Density Functions
Fußnoten
1
Readers who recollect earlier days spent on a seesaw should possess ample intuition regarding the placement of weights appropriate distances from a fulcrum so as to achieve a “balanced seesaw.”
 
2
It is an unfortunate fact of infinite sums involving both positive and negative terms that, unless the sum is absolutely convergent, an appropriate reordering of terms will result in the infinite sum converging to other real numbers (Bartle, Real Analysis, p. 292). This is hardly consistent with the notion of a balancing point of the density f(x). Moreover, the nonuniqueness of values to which the infinite sum can converge makes any particular convergence point arbitrary and meaningless as the expectation of X. Thus, to ensure the finiteness and uniqueness of the converged value in the countably infinite case, a technical condition can be added whereby E(X) is said to exist iff \( \sum\nolimits_{{x \in R(X)}} {\left| x \right|f(x) < \infty } \) which is to say, iff \( \sum\nolimits_{{x \in R(X)}} {xf(x)} \) is absolutely convergent. For virtually any problem of practical interest, if the sum used in the definition of the expectation is finite, the expectation can be said to exist. It should also be noted that in many applications, random variables are nonnegative valued, in which case if the sum is convergent, it is necessarily absolutely convergent.
 
3
The reader might notice that the mass function would exhibit properties similar to a probability density function, except the integral over the real line would not necessarily = 1, but rather equals the number reflecting the total mass placed on the rod.
 
4
Bartle, Real Analysis, pp. 213–214.
 
5
The argument can still be applied to cases where there are a finite number of discontinuities.
 
6
It is tacitly assumed that the sum and integral are absolutely convergent for the expectation to exist.
 
7
A continuous function, g, defined on a set D is called concave if \( \forall \) x∈D, \( \exists \) a line going through the point (x,g(x)) that lies on or above the graph of g. The function is convex if \( \forall \) x∈D, \( \exists \) a line going through the point (x,g(x)) that lies on or below the graph of g. The function is strictly convex or concave if the aforementioned line has only the point (x,g(x)) in common with the graph of g.
 
8
A degenerate random variable is a random variable that has one outcome that is assigned a probability of 1. More will be said about degenerate random variables in Section 3.6.
 
9
See Steven F. Arnold, (1990), Mathematical Statistics, Englewood Cliffs, NJ: Prentice Hall, pp. 92, 98.
 
10
It is tacitly assumed that the sum and integral are absolutely convergent for the expectation to exist.
 
11
Recall that the distance between the points a and b is defined by d(a,b) = |b−a| and thus squared distance would be given by d 2(a,b) = (b−a)2.
 
12
One can equivalently sum over the points (y 1,…,y m ) ∈ \( \times_{{i = 1}}^m \) R(Y i ) in defining the expectation in the discrete case.
 
13
Recall that exp(a) ≡ e a , where a is a real number.
 
14
Phoebus Dhrymes (1989) Topics in Advanced Econometrics, Springer-Verlag, p. 254.
 
15
Recall that a matrix A is positive semidefinite iff t′At ≥ 0 ∀ t, and A is positive definite iff t′At > 0 ∀ t ≠ 0.
 
16
The concept of degeneracy can be extended by calling (X 1,…,X n ) degenerate if the components satisfy one or more functional relationships (not necessarily linear) with probability 1. We will not examine this generalization here.
 
17
Equivalently, {(x,y): x = b --1(y−a), y∈R(Y)}.
 
18
For an introduction to the concept of line integrals, see E. Kreyzig (1979) Advanced Engineering Mathematics, 4th ed. New York: Wiley, Chapter 9.
 
19
See P. Billingsley (1986) Probability and Measure, 2nd ed. New York: John Wiley, pp. 73–74 for the method of proof in the discrete case.
 
20
Recall the integral inequality that if h(x) \( \geqslant \) t(x) ∀x∈(a,b), then \( \int_a^b {h(x)dx \geqslant \int_a^b {t(x)dx} } \). Strict inequality holds if h(x) > t(x) ∀x∈(a,b). The result holds for a = −∞ and/or b = ∞.
 
21
Recall that the solutions to the quadratic equation ax 2  + bx + c = 0 are given by \( x = \displaystyle \frac{{ - b\pm \sqrt {{{b^2} - 4ac}} }}{{2a}} \).
 
Metadaten
Titel
Expectations and Moments of Random Variables
verfasst von
Ron C. Mittelhammer
Copyright-Jahr
2013
Verlag
Springer New York
Buch
Mathematical Statistics for Economics and Business

Print ISBN: 978-1-4614-5021-4

Electronic ISBN: 978-1-4614-5022-1

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-1-4614-5022-1_3