76
Chapter 2 Discrete Random Variables and Probability Distributions
At this point we have considered discrete sample spaces and derived theorems con- cerning probabilities for any discrete sample space and some of the events within it.
Often, however, events are most easily described by performing some operation on the sample points. For example, if two dice are tossed, we might consider the sum showing
on the two dice; but when we find the sum, we have operated on the sample point seen. Other operations, as we will see, are commonly encountered.
We want to consider some properties of the sum; we start with the sample space. In this example a natural sample space shows the result on each die and, if the dice are
fair, leads to equally likely sample points. The sample space consists then of the 36 points in S
1
: S
1
= {.1; 1; .1; 2; : : : ; .1; 6; .2; 1; : : : ; .6; 6}: If we consider the sum on the two dice, then a sample space
S
2
= {2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12} might be considered, but now the sample points are not equally likely. We call the sum
in this example a random variable.
Definition:A random variable is a real-valued function defined on the points of a sample space.
Various functions occur commonly and we will be interested in a variety of them; sums are among the most interesting of these functions, as we will see. We will soon
determine the probabilities of various sums, but the determination of these is probably evident now to the reader. We first need, for this problem as well as for others, some
ideas and some notation.
2.1
c
Random Variables
We have considered only discrete sample spaces to this point; we discuss discrete random variables in this chapter.
First consider another example. It is convenient to let X denote the number of times an examination is attempted until it is passed. X in this case denotes a random
variable; we will use capital letters to denote random variables. We show some of the infinite sample space here, indicating the value of X; x; at each point.
Event x
P 1
F P 2
F F P 3
F F F P 4
:: :
:: :
2.1 Random Variables
77
Clearly we see that the event
′
X = 3
′
is equivalent to the event
′
F F P
′
and so their probabilities must be equal. Therefore,
P. X = 3 = P.F F P = 1
8 :
The terminology “random variable” is curious since we could, in the above ex- ample, define a variable, say Y , to be 6 regardless of the outcome of the experiment.
Y would carry no information whatsoever, and would be neither random nor variable There are other curiosities with terminology in probability theory as well, but they
have become, alas, standard in the field and so we accept them. What we call here a random variable is in reality a function whose domain is the sample space and whose
range is the real line. The random variable here, as in all cases, provides a mapping from the sample space to the real line. While being technically incorrect, the phrase
“random variable” seems to convey the correct idea. This perhaps becomes a bit more clear when we use functional notation to define a function f .x to be
f .x = P.X = x; where x denotes a value of the random variable X: In the example above we could
then write f .3 =
1 8
: The function f .x is called a probability distribution function
abbreviated as pdf for the random variable X: Since probabilities must be non-negative and since the probabilities must sum to 1,
we see that 1]
f .x ≥ 0 and
2] P
S
f .x = 1 where S is the sample space. We turn now to some examples of random variables.
Ex ample 2.1.1
Throw a fair die once and let X denote the result. The random variable X can assume the values 1, 2, 3, 4, 5, 6 and so
P. X = x = ²
1 6
for x = 1; 2; 3; 4; 5; 6 0; otherwise.
A graph of this function is of course flat; it is shown in Figure 2.1. This is an example of a discrete uniform probability distribution.
The use of a computer algebra system for sampling from this distribu- tion is explained in Appendix 1.
b
78
Chapter 2 Discrete Random Variables and Probability Distributions
Kategori