1.6. Discrete Random Variables 451.5.9.Consider an urn that contains slips of paper each with one of the num-
bers 1, 2 ,...,100 on it. Suppose there areislips with the numberi on it for
i=1, 2 ,...,100. For example, there are 25 slips of paper with the number 25. As-
sume that the slips are identical except for the numbers. Suppose one slip is drawn
at random. LetXbe the number on the slip.(a)Show thatXhas the pmfp(x)=x/ 5050 ,x=1, 2 , 3 ,...,100, zero elsewhere.(b)ComputeP(X≤50).(c)Show that the cdf ofXisF(x)=[x]([x]+1)/10100, for 1≤x≤100, where
[x] is the greatest integer inx.1.5.10.Prove parts (b) and (c) of Theorem 1.5.1.1.5.11.LetXbe a random variable with spaceD.ForD⊂D, recall that the
probability induced byXisPX(D)=P[{c:X(c)∈D}]. Show thatPX(D)isa
probability by showing the following:
(a)PX(D)=1.(b)PX(D)≥0.(c)For a sequence of sets{Dn}inD, show that{c:X(c)∈∪nDn}=∪n{c:X(c)∈Dn}.(d)Use part (c) to show that if{Dn}is sequence of mutually exclusive events,
then
PX(∪∞n=1Dn)=∑∞n=1PX(Dn).Remark 1.5.2. In a probability theory course, we would show that theσ-field
(collection of events) forDis the smallestσ-field which contains all the open intervals
of real numbers; see Exercise 1.3.24. Such a collection of events is sufficiently rich
for discrete and continuous random variables.
1.6 DiscreteRandomVariables
The first example of a random variable encountered in the last section was an
example of a discrete random variable, which is defined next.Definition 1.6.1(Discrete Random Variable). We say a random variable is a
discrete random variableif its space is either finite or countable.Example 1.6.1.Consider a sequence of independent flips of a coin, each resulting
in a head (H) or a tail (T). Moreover, on each flip, we assume that H and T are
equally likely; that is,P(H)=P(T)=^12. The sample spaceCconsists of sequences
like TTHTHHT···. Let the random variableXequal the number of flips needed