Advanced High-School Mathematics

338 CHAPTER 6 Inferential Statistics

Next, note that the limit above is a 1∞indeterminate form; taking the
natural log and applying l’Hˆopital’s rule, we have

nlim→∞

Ç

1 −

μ

n

ån

= e−μ, (6.7)

and so it follows that

P(X=k) =

e−μμk

k!

.

This gives the distribution of the Poisson random variable!

The Poisson distribution is often used to model events over time
(or space). One typical application is to model traffic accidents (per
year) at a particular intersection of two streets. For example, if our
traffic data suggests that there are roughly 2.3 accidents/year at this
intersection, then we can compute the probability that in a given year
there will be less than 2 accidents or more than 4 accidents. These
translate into the respective probabilities P(X ≤ 1) and P(X ≥ 5).
Specifically,

P(X≤1) = P(X= 0) +P(X= 1)

=

e−^2.^32. 30

0!

+

e−^2.^32. 31

1!

= e−^2.^3 (1 + 2.3) = 3. 3 e−^2.^3 ≈. 331.

In the same vein,

P(X≥5) = 1−P(X≤4)

= 1−

Çe− 2. (^32). 30
0!
+e
− 2. (^32). 31
1!
+e
− 2. (^32). 32
2!
+e
− 2. (^32). 33
3!
+e
− 2. (^32). 34
4!
å
= 1−e−^2.^3
Ç
1 + 2.3 +^2.^3
2
2!
+^2.^3
3
3!
+^2.^3
4
4!
å
≈ 0. 081.