Poisson distribution [Miller n Freund] Matlab [Martinez]

Poisson Distribution is approximation for Binomial Distribution when n --> inf and p --> 0, smalll ( so np is moderate).
Derived from Binomial Dist Eqn by sub var p with λ/n .

where λ= np
Expected value E[X] = λ and variance V(X) = λ (replace np=λ p-->0)

Example:

5% of of bounded book at certain bindery centre have defective. Find the probability that 2 of 100 books bounded by this bindery centre is defective:












Poisson Process:

Extending the uses of above formula for process taking place over continuous interval of time. i.e: Events occur at points in time or space

To find the probability of x success during a time interval of length T, we devided the interval T into n equal parts of length ∆t, with the probability of success p = α ∆t. α is the average (mean) of successes per unit time.

Assumption:

1. The probability of a success during a very small interval, ∆t, is given by p = α ∆t.
2. The probability of > one success during such a small time interval ∆t is negligible.
3. The probability of a success during such a time interval does not depend on what happened prior to that time.
   

The formula for Poisson distribution can be futher expand by expading the parameter λ.
λ = n.p = (T/∆t) *(α∆t) = αT

Note: However most of the time we use symbol λ to represent α .

Example:

Bank receives average λ= 6 bad checks per day, what are the probabilty that it will receive:
a: 4 bad checks on any given day.

• f(x;λT) = f(4;6(1))=0.135
• MATLAB: prob = poisspdf(4,6) = 0.1339
• or prob = poisscdf(4,6)-poisscdf(3,6) = 0.1339
  

b: at most 10 bad checks on any two consecutive day.

• f(x;λT) = f(x;6(2))=f(0;12) + f(1; 12)+ ...+ f(10;12)= 0.347
• MATLAB: prob = sum(poisspdf(0:10,12) = 0.3472
• or prob = poisscdf(10,12) = 0.3472