for applied maths

1. Presentation
on
Poisson Distribution-Assumption , Mean &
Variance

2.  A discrete Probability Distribution
 Derived by French mathematician Simeon Denis
Poisson in 1837
 Defined by the mean number of occurrences in a time
interval and denoted by λ
 Also known as the Distribution of Rare Events
Poisson Distribution
Simeon D. Poisson (1781-
1840)

3. Works when binomial calculation becomes impractical (No. of
trials>probability of success),
 Applied where random events in space or time are expected to occur.
 Deviation indicates some degree of non-randomness in the events
Example: Number of earthquakes per year.
Cont’d…

4. Requirements for a Poisson Distribution
RIPS
Random
Proportional
Simultaneous
Independent

5. Assumptions
 The probability of occurrence of an event is constant for
all subintervals:
 There can be no more than one occurrence in each
interval
 Occurrence are independent .

6. If the Poisson variable X, then by the formula: P(X = x) = e-l
lx
x!
Mean and Variance
Mean of Poisson Probability Distribution
Variance of Poisson Probability Distribution
Or λ = np

7. Mathematical Calculations
#If the average number of accidents at a particular intersection in
every year is 18. Then-
(a) Calculate the probability that there are exactly 2 accidents
occurred in this month.
(b) Calculate the probability that there is at least one accident
occurred in this month.

8. There are 12 months in a year, so l = 12
18
= 1.5 accidents per month
P(X = 3) =
!
x
e x
l
l
-
!
2
5
.
1 2
5
.
1
-

e
= 0.2510
(a) Calculate the probability that there are exactly 2
accidents occurred in this month.

9. (b) Calculate the probability that there is at least one accident
occurred in this month.
P(X ≥ 1 ) = P(X=1) + P(X=2) + P(X=3) + …. Infinite.
So… Take the complement: P(X=0)
!
x
e x
l
l
-

!
0
5
.
1 0
5
.
1
-

e
5
.
1
-
 e
= 0.223130…