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 5.2 shoplifters per day. (A day consists of a 12-hour period.) What is the probability that Market Mall security catches
a) ten shoplifters a day?_____
b) at least three shoplifters a day?____
c)Once a shoplifter has been caught, how much time would expect to pass until the next shoplifter is caught?___
Standard deviation____

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