Consider the probability function
For many probability functions there exists a function
Given a probability function
A function
It is necessary that
A function
Some typical probability density functions are illustrated in Fig.2A .
Example 2A . Verifying that a function is a probability density function . Suppose one is told that the time one has to wait for a bus on a certain street corner is a numerical-valued random phenomenon, with a probability function, specified by the probability density function
The function
The function
Example 2B . Computing probabilities from a probability density function . Let us consider again the numerical-valued random phenomenon, discussed in example 1A , that consists in observing the time one has to wait for a bus at a certain bus stop. Let us assume that the probability function
From (2.1) it follows that if
Example 2C . The lifetime of a vacuum tube. Consider the numerical valued random phenomenon that consists in observing the total time a vacuum tube will burn from the moment it is first put into service. Suppose that the probability function
Let
For many probability functions there exists a function
Given a probability function
A function
Example 2D . Computing probabilities from a probability mass function . Let us consider again the numerical-valued random phenomenon considered in examples 1A and 2B. Let us assume that the probability function
An algebraic formula for
It then follows that
The terminology of “density function” and “mass function” comes from the following physical representation of the probability function
We shall see in section 3 that a probability function
Exercises
Verify that each of the functions
2.1.
2.2.
2.3.
2.4.
2.5.
Show that each of the functions
Hint : use freely the facts developed in the appendix to this section.
2.6.
2.7.
2.8. The amount of bread (in hundreds of pounds) that a certain bakery is able to sell in a day is found to be a numerical-valued random phenomenon, with a probability function specified by the probability density function
- Find the value of
which makes a probability density function. - Graph the probability density function.
- What is the probability that the number of pounds of bread that will be sold tomorrow is (a) more than 500 pounds,
less than 500 pounds, (c) between 250 and 750 pounds? - Denote, respectively, by
, and , the events that the number of pounds of bread sold in a day is ( ) greater than 500 pounds, less than 500 pounds, between 250 and 750 pounds. Find . Are and independent events? Are and independent events?
2.9. The length of time (in minutes) that a certain young lady speaks on the telephone is found to be a random phenomenon, with a probability function specified by the probability density function
(i) Find the value of
(ii) Graph the probability density function.
(iii) What is the probability that the number of minutes that the young lady will talk on the telephone is
(iv) For any real number
Answer
(i)
2.10. The number of newspapers that a certain newsboy is able to sell in a day is found to be a numerical-valued random phenomenon, with a probability function specified by the probability mass function
(i) Find the value of
(ii) Sketch the probability mass function.
(iii) What is the probability that the number of newspapers that will be sold tomorrow is (a) more than 50, (b) less than 50, (c) equal to 50, (d) between 25 and 75, inclusive, (e) an odd number?
(iv) Denote, respectively, by
2.11. The number of times that a certain piece of equipment (say, a light switch) operates before having to be discarded is found to be a random phenomenon, with a probability function specified by the probability mass function
(i) Find the value of
(ii) Sketch the probability mass function.
(iii) What is the probability that the number of times the equipment will operate before having to be discarded is (a) greater than
(iv) For any real number
Answer
(i)
Appendix: The Evaluation of Integrals and Sums
If (2.1) and (2.7) are to be useful expressions for evaluating the probability of an event, then techniques must be available for evaluating sums and integrals. The purpose of this appendix is to state some of the notions and formulas with which the student should become familiar and to collect some important formulas that the reader should learn to use, even if he lacks the mathematical background to justify them.
To begin with, let us note the following principle. If a function is defined by different analytic expressions over various regions, then to evaluate an integral whose integrand is this function one must express the integral as a sum of integrals corresponding to the different regions of definition of the function . For example, consider the probability density function
An important integration formula, obtained by integration by parts, is the following, for any real number
We next consider the Gamma function
The Gamma function is a generalization of the factorial function in the following sense. From (2.13) it follows that
Next, it may be shown that for any integer
We prove (2.21) by showing that
In view of (2.22) , to establish (2.21) we need only show that
We prove (2.23) by proving the following basic formula; for any
Equation (2.24) may be derived as follows. Let
We now evaluate the double integral in (2.25) by means of a change of variables to polar coordinates. Then
For large values of
Take next
We obtain an important generalization of the binomial theorem by taking
For the case of
Equation (2.34) , with
Equation (2.34) with
From (2.33) we may obtain another important formula. By a comparison of the coefficients of
If
Theoretical Exercises
2.1 . Show that for any positive real numbers
2.2 . Show for any
2.3 . The integral
Show finally that the beta and gamma functions are connected by the relation
2.5 . Prove that the integral defining the gamma function converges for any real number
2.6 . Prove that the integral defining the beta function converges for any real numbers
2.7 . Taylor’s theorem with remainder . Show that if the function
2.8 . Lagrange’s form of the remainder in Taylor’s theorem . Show that if
- We usually assume that the integral in (2.1) is defined in the sense of Riemann; to ensure that this is the case, we require that the function
be defined and continuous at all but a finite number of points. The integral in (2.1) is then defined only for events , which are either intervals or unions of a finite number of non-overlapping intervals. In advanced probability theory the integral in (2.1) is defined by means of a theory of integration developed in the early 1900’s by Henri Lebesgue. The function must then be a Borel function, by which is meant that for any real number the set is a Borel set. A function that is continuous at all but a finite number of points may be shown to be a Borel function. It may be shown that if a Borel function satisfies (2.1) and (2.3) then, for any Borel set , the integral of over exists as an integral defined in the sense of Lebesgue. If is an interval, or a union of a finite number of non-overlapping intervals, and if is continuous on , then the integral of over , defined in the sense of Lebesgue, has the same value as the integral of over , defined in the sense of Riemann. Henceforth, in this book the word function (unless otherwise qualified) will mean a Borel function and the word set (of real numbers) will mean a Borel set. ↩︎ - For the purposes of this book we also require that a probability density function
be defined and continuous at all but a finite number of points. ↩︎ - The reader should note the convention used in the exercises of this book. When a function
is defined by a single analytic expression for all in , the fact that varies between and is not explicitly indicated. ↩︎
