Chapter 5 Probability Distributions 185
Discrete Probability Distributions
In a discrete probability distribution, the probabilities are associated with
a series of discrete outcomes. The probabilities associated with tossing a
coin form a discrete distribution since there are two separate and distinct
outcomes. If you toss a 6-sided die, the probabilities associated with that out-
come also form a discrete distribution, where each side has a^16 probability of
turning up. We can write this as
p^1 y^25
1
6
; y 5 1, 2, 3, 4, 5, 6
where p(y) means the “probability of y,” for integer values of y ranging from
1 to 6.
Note that discrete does not mean “fi nite.” There are discrete probability
distributions that cover an infi nite number of possible outcomes. One of
these is the Poisson distribution, used when the outcome event involves
counts within a specifi ed period of time. The equation for the Poisson dis-
tribution is
Poisson Distribution
p^1 y^25
ly
y!
e2l^ y 5 0,1, 2,c
where l (pronounced “lambda”) is the average number of events in the
specifi ed time period and y! stands for “y factorial,” which is equal to the
product y(y 2 1)(y 2 2) ... (3)(2)(1). For example, 4! 5 4 3 3 3 2 3 1 5 24.
Lambda is an example of a parameter, a term in the formula for a probabil-
ity distribution that defi nes its shape and values. Let’s see what probabilities
are generated for a specifi c value of l.
Suppose we want to determine the number of car accidents at an intersec-
tion in a given year and we know that the average number of accidents is 3.
What is the probability of exactly two accidents occurring that year? The
Poisson distribution usually applies to this situation. In this case, the value of
l is 3, y 5 2, and the probability is
32
2!
e^235
9 #0. 0498
2 # 1
5 0.224
or the probability of exactly two accidents occurring at the intersection is
about 22%. Note that the probabilities extend across an infi nite number of
possible integer values.
Discrete distributions can be displayed with a bar chart in which the
height of each bar is proportional to the probability of the event. Figure 5-2
displays the probability distribution from y 5 0 to y 5 10 accidents per year.
