Remarks on more general distributions

14.15 Remarks on more general distributions

In the foregoing sections we have discussed examples of discrete and continuous distribu- tions. The values of a discrete distribution are computed by adding the values of the corresponding probability mass function. The values of a continuous distribution with a density are computed by integrating the density function. There are, of course, distribu- tions that are neither discrete nor continuous. Among these are the so-called “mixed”

types in which the mass distribution is partly discrete and partly continuous. (An example is shown in Figure 14.3.)

A distribution function F is called mixed if it can be expressed as a linear combination of the form

where is discrete and

must satisfy the relations 0<

is continuous. The constants

and

Properties of mixed distributions may be found by studying those that are discrete or continuous and then appealing to the linearity expressed in Equation (14.26).

A general kind of integral, known as the Riemann-Stieltjes integral, makes possible a simultaneous treatment of the discrete, continuous, and mixed cases.7 Although this integral unifies the theoretical discussion of distribution functions, in any specific problem the computation of probabilities must be reduced to ordinary summation and integration. In this introductory account we shall not attempt to describe the Riemann-Stieltjes integral. Consequently, most of the topics we discuss come in pairs, one for the discrete case and one for the continuous case. However, we shall only give complete details for one case, leaving the untreated case for the reader to work out.

Even the Riemann-Stieltjes integral is inadequate for treating the most general distribu- tion functions. But a more powerful concept, called the Lebesgue-Stieltjes

does give a satisfactory treatment of all cases. The advanced theory of probability cannot be undertaken without a knowledge of the Lebesgue-Stieltjes integral.

A discussion of the Riemann-Stieltjes integral may be found in Chapter 9 of the author’s Mathematical Analysis, Addison-Wesley Publishing Company, Reading, Mass. 1957. See any book on measure theory.

Calculus of probabilities

14.16 Exercises

1. Let X be a random variable which measures the lifetime (in hours) of a certain type of vacuum

tube. Assume X has an exponential distribution with parameter = 0.001 . Determine T

so that > is (a) 0.90; (b) 0.99. You may use the approximate formula -log (1 x) = x+

in your calculations.

2. A radioactive material obeys an exponential decay law with half-life 2 years. Consider the decay time X (in years) of a single atom and assume that X is a random variable with an exponential distribution. Calculate the probability that an atom disintegrates (a) in the interval

1 X 2 (b) in the interval 2 X 3 ; (c) in the interval 2 X 3, given that it has not disintegrated in the interval 0 X 2 ; (d) in the interval 2 X 3, given that it has not disintegrated in the interval 1 X 2.

3. The length of time (in minutes) of long distance telephone calls from is found to be a random phenomenon with probability density function

for

0 for

Determine the value of c and calculate the probability that a long distance call will last (a) less than 3 minutes; (b) more than 6 minutes; (c) between 3 and 6 minutes; (d) more than

9 minutes.

4. Given real constants > 0 and c. Let

if

if <c.

Verify that dt = 1, and determine a distribution function F havingfas its density.

This is called an exponential distribution with two parameters, a decay parameter

and a

location parameter

5. State and prove an extension of Theorem 14.9 for exponential distributions with two parameters and c.

6. A random variable has an exponential distribution with two parameters and c. Let Y = +

where a > 0. Prove that Y also has an exponential distribution with two parameters I’ and

and determine these parameters in terms of a, c, and ii.

7. In Exercise 16 of Section 11.28 it was shown that

Use this result to prove that for > 0 we have

dx =

8. A random variable X has a standard normal distribution Prove that (a) =1 < k) =

1 > k) =

Q(k)).

9. A random variable X has a standard normal distribution Use Table 14.1 to calculate each of the following probabilities: (a)

10. A random variable X has a standard normal distribution Use Table 14.1 to find a number c such that

11. Assume Xhas a normal distribution function Fwith mean and variance and let denote the standard normal distribution.

Distributions of functions of random 541 (a) Prove that

(b) Find a value of c such that

(c) Find a value of c such that

> c) = 0.98.

12. A random variable Xis normally distributed with mean m = and variance = 4. Calculate each of the following probabilities: (a)

-3 3) ; (b)

13. An architect is designing a doorway for a public building to be used by people whose heights are normally distributed, with mean m = 5 ft. 9 in., and variance where = 3 in. How low

can the doorway be so that no more than 1 of the people bump their heads?

14. If X has a standard normal distribution, prove that the random variable Y = + is also normal if a 0. Determine the mean and variance of Y.

15. Assume a random variable has a standard normal distribution, and let Y = (a) Show that

if

(b) Determine when < 0 and describe the density .