Two random variables,
The joint probability function, denoted by
in which
instead of (5.1). However, it should be kept constantly in mind that the right-hand side of (5.2) is without mathematical content of its own; rather, it is an intuitively meaningful concise way of writing the right-hand side of (5.1) .
It is useful to think of the joint probability function
We are particularly interested in knowing the value
In order to know the joint probability function
as the set consisting of all 2-tuples
In words,
In terms of the probability mass distributed over the plane of Fig. 7A of Chapter 4,
The reader should verify for himself the following important formula [compare (7.3) of Chapter 4]: for any real numbers
It is important to note that from a knowledge of the joint distribution function
Similarly, for any real number
In terms of the probability mass distributed over the plane by the joint distribution function
The function
We next define the joint probability mass function of two random variables
It may be shown that there is only a finite or countably infinite number of 2-tuples
Two jointly distributed random variables,
Two jointly distributed random variables,
By letting
Next, for any real numbers
The joint probability density function may be obtained from the joint distribution function by routine differentiation, since
at all 2-tuples
If the random variables
The reader should compare (5.14) with (3.4) .
To prove (5.14), one uses the fact that by (5.6), (5.7), and (5.11),
The foregoing notions extend at once to the case of
The joint probability mass function is given by
A continuous joint probability law is specified by its probability density function: for any Borel set
The individual (or marginal ) probability law of each of the random variables
An analogous formula may be written in the discrete case for
Example 5A . Jointly discrete random variables . Consider a sample of size 2 drawn with replacement (without replacement) from an urn containing two white, one black, and two red balls. Let the random variables
Solution
The random variables
Example 5B . Jointly continuous random variables . Suppose that at two points in a room (or on a city street or in the ocean) one measures the intensity of sound caused by general background noise. Let
Find the individual probability density functions of
Solution
By (5.14) , the individual probability density functions are given by
Note that the random variables
The probability that the sum of the sound intensities is less than 1 is given by
Example 5C . The maximum noise intensity . Suppose that at five points in the ocean one measures the intensity of sound caused by general background noise (the so-called ambient noise). Let
Define
Theoretical Exercise
5.1 . Multivariate distributions with given marginal distributions . Let
in which
Equations (5.22) and (5.23) are due to E. J. Gumbel, “Distributions à plusieurs variables dont les marges sont données”,
Exercises
In exercises 5.1 to 5.3 consider a sample of size 3 drawn with replacement (without replacement) from an urn containing (i) 1 white and 2 black balls,
(ii) 1 white, 1 black, and 1 red ball. For
5.1 . Describe the joint probability law of
Answer
| (i), (ii) | ||
|---|---|---|
| with | ||
| otherwise | 0; | |
| without, | ||
| otherwise | 0. |
5.2 . Describe the individual (marginal) probability laws of
5.3 . Describe the individual probability laws of the random variables
Answer
With,
without,
In exercises 5.4 to 5.6 consider 2 random variables,
(a)
(b)
5.4 . Find (i)
5.5 . Find (i)
Answer
(a) (i)
5.6 . Find (i)
In exercises 5.7 to 5.10 consider 2 random variables,
5.7 . Show that the individual probability mass functions of
Answer
Yes.
5.8 . Find (i)
5.9 . Find (i)
Answer
(a) (i)
5.10 . Find (i)
