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62 Chapter 2 Review of Probability

Probability Distribution of a Discrete
Random Variable

Probability distribution. The probability distribution of a discrete random vari-
able is the list of all possible values of the variable and the probability that each
value will occur. These probabilities sum to 1.

For example, let M be the number of times your computer crashes while you
are writing a term paper. The probability distribution of the random variable M
is the list of probabilities of each possible outcome: The probability that M = 0,
denoted Pr(M = 0), is the probability of no computer crashes; Pr(M = 1) is the
probability of a single computer crash; and so forth. An example of a probability
distribution for M is given in the second row of Table 2.1; in this distribution, if
your computer crashes four times, you will quit and write the paper by hand.
According to this distribution, the probability of no crashes is 80%; the probabil-
ity of one crash is 10%; and the probability of two, three, or four crashes is,
respectively, 6%, 3%, and 1%. These probabilities sum to 100%. This probability
distribution is plotted in Figure 2.1.

Probabilities of events. The probability of an event can be computed from
the probability distribution. For example, the probability of the event of one or
two crashes is the sum of the probabilities of the constituent outcomes. That
is, Pr(M = 1 or M = 2) = Pr(M = 1) + Pr(M = 2) = 0.10 + 0.06 = 0.16, or
16%.

Cumulative probability distribution. The cumulative probability distribution
is the probability that the random variable is less than or equal to a particular
value. The last row of Table 2.1 gives the cumulative probability distribution of
the random variable M. For example, the probability of at most one crash,
Pr(M … 1), is 90%, which is the sum of the probabilities of no crashes (80%) and
of one crash (10%).

TABLE 2.1 Probability of Your Computer Crashing M Times

Outcome (number of crashes)

012 3 4
0.01
Probability distribution 0.80 0.10 0.06 0.03
1.00
Cumulative probability 0.80 0.90 0.96 0.99
distribution