Distribution Function

Definition 3.1.5. The distribution function F(x) of a random variable X(a>) is defined by

F(x) = P{o)Х(оз) < x).

Note that the distribution function can be defined for any random variable because a probability is assigned to every element of A and hence to {cojX(eo) < x) for any x. We shall write P{o)X(oj) < x} more compactly as P(X<x).

A distribution function has the properties:

(i) F(-«) = 0.

(ii) FM=1.

(iii) It is nondecreasing and continuous from the left.

[Some authors define the distribution function as F(x) = P{toX(a>) ё x}. Then it is continuous from the right.]

Using a distribution function, we can define the expected value of a random variable whether it is discrete, continuous, or a mixture of the two. This is done by means of the Riemann-Stieltjes integral, which is a simple generaliza­tion of the familiar Riemann integral. Let X be a random variable with a distribution function F and let Y = h(X), where A( •) is Borel-measurable.6 We define the expected value of Y, denoted by EY as follows. Divide an interval [a, b] into n intervals with the end points a — x„ < x, < . . . < x„_! < x„ = b and let x f be an arbitrary point in [x,, x,+, ]. Define the partial sum

= 2 h{xf )[F(xi+1) – F(xt)] (3.1.1)

1-0

associated with this partition oftheinterval [a, b]. If, for any e > 0, there exists a real number A and a partition such that for every finer partition and for any choice of xf, IS’n — AI < e, we call A the Riemann-Stieltjes integral and denote it by jbah{x) dF(x). It exists if A is a continuous function except possibly for a countable number of discontinuities, provided that, whenever its discontinu­
ity coincides with that of F, it is continuous from the right.7 Finally, we define

image174(3.1.2)

provided the limit (which may be +°° or —°°) exists regardless of the way a —*—oo and b-*<*>.

If dF/dxexists and is equal to /(x), F(xi+l) — F(x,) — /(x*)(xm — x,) for some x? Є [xt+i, x,] by the mean value theorem. Therefore

image175(3.1.3)

On the other hand, suppose X= ct with probability ph і =1,2,. . . , К. Take a < c, and cK < t, then, for sufficiently large n, each interval contains at most one of the cjs. Then, of the n terms in the summand of (3.1.1), only К terms containing cjs are nonzero. Therefore

image176(3.1.4)

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