ABSTRACT
As discussed in Section 4.2, the distribution function of a random variable X is a function F from (−∞, +∞) to R defined by F (t) = P (X ≤ t). From the definition of F we deduced that it is nondecreasing, right continuous, and satisfies limt→∞ F (t) = 1 and limt→−∞ F (t) = 0. Furthermore, we showed that, for discrete random variables, distributions are step functions. We also proved that if X is a discrete random variable with set of possible values {x1, x2, . . .}, probability mass function p, and distribution function F , then F has jump discontinuities at x1, x2, . . . , where the magnitude of the jump at xi is p(xi) and for xn−1 ≤ t < xn,
F (t) = P (X ≤ t) = n−1∑
p(xi). (6.1)
In the case of discrete random variables a very small change in t may cause relatively large changes in the values of F . For example, if t changes from xn − ε to xn, ε > 0 being arbitrarily small, then F (x) changes from
∑n−1 i=1 p(xi) to
∑n i=1 p(xi), a change of
magnitude p(xn), which might be large. In cases such as the lifetime of a random light bulb, the arrival time of a train at a station, and the weight of a random watermelon grown in a certain field, where the set of possible values of X is uncountable, small changes in x produce correspondingly small changes in the distribution of X . In such cases we expect that F , the distribution function of X , will be a continuous function. Random variables that have continuous distributions can be studied under general conditions. However, for practical reasons and mathematical simplicity, we restrict ourselves to a class of random variables that are called absolutely continuous and are defined as follows:
Definition Let X be a random variable. Suppose that there exists a nonnegative realvalued function f : R → [0,∞) such that for any subset of real numbers A that can be constructed from intervals by a countable number of set operations,
P (X ∈ A) = ∫
f(x) dx. (6.2)
Then X is called absolutely continuous or, in this book, for simplicity, continuous. Therefore, whenever we say that X is continuous, we mean that it is absolutely continuous and hence satisfies (6.2). The function f is called the probability density function, or simply the density function of X .
