1.5. Random Variables 37light smokers were five and three times that of the nonsmokers, respectively. A ran-
domly selected participant died over the five-year period: calculate the probability
that the participant was a nonsmoker.
1.4.34.A chemist wishes to detect an impurity in a certain compound that she is
making. There is a test that detects an impurity with probability 0.90; however,
this test indicates that an impurity is there when it is not about 5% of the time.
The chemist produces compounds with the impurity about 20% of the time. A
compound is selected at random from the chemist’s output. The test indicates that
an impurity is present. What is the conditional probability that the compound
actually has the impurity?1.5 RandomVariables
The reader perceives that a sample spaceCmay be tedious to describe if the elements
ofCare not numbers. We now discuss how we may formulate a rule, or a set of
rules, by which the elementscofCmay be represented by numbers. We begin the
discussion with a very simple example. Let the random experiment be the toss of
a coin and let the sample space associated with the experiment beC={H, T},
whereHandTrepresent heads and tails, respectively. LetXbe a function such
thatX(T)=0andX(H)=1. ThusXis a real-valued function defined on the
sample spaceCwhich takes us from the sample spaceCto a space of real numbers
D={ 0 , 1 }. We now formulate the definition of a random variable and its space.
Definition 1.5.1. Consider a random experiment with a sample spaceC.Afunc-
tionX, which assigns to each elementc∈Cone and only one numberX(c)=x,is
called arandom variable.ThespaceorrangeofXis the set of real numbers
D={x:x=X(c),c∈C}.
In this text,Dgenerally is a countable set or an interval of real numbers. We call
random variables of the first typediscreterandom variables, while we call those of
the second typecontinuousrandom variables. In this section, we present examples
of discrete and continuous random variables and then in the next two sections we
discuss them separately.
Given a random variableX,itsrangeDbecomes the sample space of interest.
Besides inducing the sample spaceD,Xalso induces a probability which we call
thedistributionofX.
Consider first the case whereXis a discrete random variable with a finite space
D={d 1 ,...,dm}. The only events of interest in the new sample spaceDare subsets
ofD. The induced probability distribution ofXis also clear. Define the function
pX(di)onDby
pX(di)=P[{c:X(c)=di}], fori=1,...,m. (1.5.1)
In the next section, we formally definepX(di)astheprobability mass function
(pmf)ofX. Then the induced probability distribution,PX(·), ofXis
PX(D)=∑di∈DpX(di),D⊂D.