Question

IN AN EXCEL FILE USING EXCEL FUNCTIONS CALCULATIONS SHOW FORMULAS IN CELL PLEASE The annual number of industrial accidents occurring in a manufacturing plant is known to follow a Poisson distribution with mean 12. a. What is the probability of observing of observing exactly 12 accidents during the coming year? b. What is the probability of observing no more than 12 accidents during the coming year? c. What is the probability of observing at least 15 accidents during the coming year? d. What is the probability of observing between 10 and 15 accidents (including 10 and 15) during 1) the coming year? Find the smallest integer k such that we can be at least 99% sure that the annual number of accidents occurring will be less than k. e.

Answer 1

Let X ~ Poi( = 12 )

a) Here we want to find P( X = 12)

Let's use excel:

P(X = 12) = "=POISSON(12,12,0)" = 0.1144

b) Here we want to find P( X <= 12) = "=POISSON(12,12,1)" = 0.5760

c) Here we want to find P( X >= 15) = 1 - P( X <= 14) ...( 1 )

Let P(X <= 14) = "=POISSON(14,12,1)" = 0.7720

Plug this value in equation ( 1 ), we get.

P(X >= 15) = 1 - 0.7720 = 0.2280

d) Here we want to find P( 10 <= X <= 15 ) = P(X <= 15) - P(X <= 9) ......( 2 )

P(X <= 15 ) = "=POISSON(15,12,1)" = 0.8444

P(X <= 9 ) = "=POISSON(9,12,1)" = 0.2424

Plug these values in equation ( 1 ), we get.

P( 10 <= X <= 15 ) = 0.8444 - 0.2424 = 0.6020

e) Here we want to find k such that P(X < k) >= 0.99

Let's make cumulative Poisson probability table in excel:

The formulae used on the above excel-sheet are as follows:

Therefore answer of part e) is k = 21