POISSON distribution background.
Napoleon
Study of deaths of soldiers from horse-kicks. The probability distribution is
Poisson. It has only one parameter, lambda= the average of something.
The mean and variance of a Poisson random variable is lambda. It is the simplest probability distribution to work with in the sense that it needs little
data. (Only the average value is needed). Bell Labs designed US telephone network by using the Poisson model. The range of x for Poisson is 0 to infinity.
Poisson Formula:
P(X=x)= e^(-lambda) * (lambda^x) / x!
Poisson Dist. Example
Cars enter a car wash (like a Poisson) at a rate of 4 per half hour.
Find the prob. that there may be more than four cars entering in an hour.
Solution:
lambda=average per hour=8,
lambda is the only parameter of the Poisson distribution
range of x for Poisson is 0 to infinity (always the same range)
Probability of the Poisson variable when x is more than 4 is Asked from the wording of the problem. This may be tedious calculation. We need to become smart Alecs and compute what is NOT ASKED and then get what it ASKED by
subtracting our answer for what is NOT ASKED from 1
Remember the Complement Rule for probability P(A)+ P(Complement of A) =1
P(WHAT IS ASKED) + P(WHAT IS NOT ASKED) = 1
otherwise, it will not be a probability distribution.
x=0 is Not Asked ( i.e., this x is not wanted by this problem)
(Why? 0 is NOT more than 4, it is less than 4)
x=1 is NOT Asked
x=2 is NOt Asked
x=3 is NOt Asked
x=4 is Not Asked ( they want MORE than 4, 4 is not more than 4)
x=5 is Asked
x=6 is Asked
x=500 is Asked because 500 IS MORE THAN 4
HERE are TOO many Asked's to compute with No end in sight
We do know that the total probability of all x's from 0 to infinity
is 1 ( a property of any probability distribution, Poisson is one such
prob. dist'n)
So we will compute prob for x=0 to x=4 (the NOT Asked's)
Add them up
Sum= P(x=0)+P(x=1)+ P(x=2) + P(x=3)+ P(x=4)
and subtract the added up sum from 1
FINAL ANSWER is 1 MINUS Sum
In computing these P(x)= e^(- lambda) is first part
lambda ^ x is 2nd part
x! is 3rd part to go in denominator
P(x) = 1st part times 2nd part divided by the 3rd part
1st part remains the same for each x because there is Not Askd x in
this part
So i suggest computing it only once and storing it in your calculator.
Learn touse the STO button and RCL (recall) button
some calculators have Min for memory in and Mrc for memory recall
e^(-8) is 0.000335462 this is P(0)=0.0003 rounded to 4 places
0.00268370 this is P(1)=0.0027
0.010734804 this is P(2)=0.0107
0.028626 this is P(3)=0.0286
0.057252288 this is P(4)=0.0573
SUM of these=0.0996 = P(Not Asked)
Than answer is 1-P(NOT ASKED) = 0.9004
A TRICK USEFUL in SEQUENTIAL COMPUTATION of POISSON probabilities.
What is sequential computation?
example: probabilities from x=3 to x=8 in a sequence.
The trick saves time in keying into the calculator (e^-lambda).
Some may find that learning the trick is more trouble than it is worth! I think it is fun.
If we have computed P(X=a) and want to compute P(X=a+1)
it is simply obtained by multiplying P(X=a) by lambda and dividing by (a+1)
For example, let a=3, a+1=4. Now the trick says how to get
P(x=4) from P(x=3)
in order
to get P(X=a+1)=P(x=4) from P(X=a)=P(x=3)=0.028626
we simply multiply 0.028626*lambda/(a+1), that is,
we simply multiply 0.028626*8/(3+1) or 0.57252