The number of shoplifters caught by security at Market Mall

The number of shoplifters caught by security at Market Mall is a random variable which can be modeled using the Poisson distribution, with an average of 6.2 shoplifters per day. (A day consists of a 12-hour period.) What is the probability that Market Mall security catches (Input answers to four decimal places)

(a) ten shoplifters a day?

(b) at most three shoplifters a day?

(c) Once a shoplifter has been caught, how much time would you expect to pass until the next shoplifter is caught?

Provide the standard deviation as well.

Solution

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a)
P( X = 10 ) = e ^-6.2 * 6.2^10 / 10! = 0.0469

b)
P( X < = 3) = P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= e^-6.2 * 6.2 ^ 3 / 3! + e^-6.2 * 10 ^ 2 / 2! + e^-6.2 * ^ 1 / 1! + e^-6.2 * ^ 0 / 0!
= 0.1342