Question

The number of letters arriving each day at a residential address is assumed to be Poisson distributed with mean 1.8. The numbers of letters arriving on different days are independent random variables. (i) Calculate the probability that exactly two letters arrive at the address in one day. (ii) Calculate the probability that no more than 5 letters arrive at this address in a 5 day period. (iii) On a particular day, there are no letters at this address. Find the probability that exactly 6 days go by before this happens again. (iv) Use a suitable approximation to calculate the probability that during a 30 day period, more than 65 letters are received at this address, with mean rate λ = 1.8 for each day

Answer 1

The number of letters arriving each day at a residential address has Poisson distribution.

The Poisson PMF is .

a)The probability that exactly two letters arrive at the address in one day is

b) The Poisson parameter (mean) for 5 day period is .

The probability that no more than 5 letters arrive at this address in a 5 day period is

Use R. The command is

> ppois(5,9)
[1] 0.1156905

iii) The probability that no letters arrive on a particular day is

The probability that at least one letter arrive on a particular day is

.

iii) The probability of the event " No letters on the first day. At least one letter on the next 6 days. No letters on the eight day "

is (Since The numbers of letters arriving on different days are independent random variables)

iv) The sum of 30 Poisson random variables has approximately Normal distribution (Central Limit Theorem) with  

The probability that more than 65 letters are received at this address is

The exact probability is

> 1-ppois(65,54)
[1] 0.06229324