Question

The number of female customers arriving to a coffee shop follow a Poisson process with a mean rate of 3 per hour. The number of male customers arriving to the same coffee shop also follow a Poisson process with a mean rate of 6 per hour and their arrival is independent of the arrivals of female customers.

a) What is the probability that the next customer will arrive within 5 minutes?

b) What is the probability that exactly thee customers will arrive in the next 5 minutes?

c) What is the probability of exactly two male and exactly one female customer will arrive in the next 5 minutes?

d). Parts b) and c) ask for the probability of exactly three arrivals in the next 5 minutes. Are they identical? Explain why?

e) What is the probability of exactly five customers will arrive between 6 and 7 hours from now?

Answer 1

a)mean rate for female = 3per hour = 3/12 per 5 minutes = 1/4 per 5 minutes

mean rate for male =6 per hour = 6/12 per 5 minutes = 1/2 per 5 minutes

a)Probablity tha female customer arrive in 5 minutes = 1-P(0) = 1- e^-1/4 =1- 0.7788 = 0.2212

Probablity tha male customer arrive in 5 minutes = 1-P(0) = 1- e^-1/2 =1- 0.60653 = 0.39347

Probabilty that next customer arrive within 5 minutes = 0.2212+0.39347 -0.2212*0.39347 = 0.5276

b)Probabilty that exactly three customers will in next 5 minutes

= P(F = 0)*P(M=3)+ P(F = 1)*P(M=2)+ P(F = 2)*P(M=1)+ P(F = 3)*P(M=0)

P(F=0) = e^-1/4 .= 0.7788 ,P(M=3) = e^-1/2(1/2)³/3! = 0.0126

P(F=1) = e^-1/4(1/4). = 0.1947 ,P(M=2) = e^-1/2(1/2)²/2! = 0.0758

P(F=2) = e^-1/4(1/4)²/2!. = 0.0243 ,P(M=1) = e^-1/2(1/2)¹/1! =0.3032

P(F=3) = e^-1/4(1/4)³/3! = 0.0020 ,P(M=0) = e^-1/2 = 0.60653

Probabilty that exactly three customers will in next 5 minutes

= 0.7788*0.0126+0.1947*0.0758+0.0243*0.3032+0.0020*0.60653 = 0.03315196

e)

as both arrival of male and female are independs ; expected number of customers in one hour

=expected male+expected female =3+6 =9 =

hence from poisson distribution  probability of exactly five customers will arrive between 6 and 7 hours from now

=P(X=5) =e^-9*9⁵/5! =0.0607