I have used excel formula = poisson.dist. Where x = number of events, mean = mean, and cumulative = True/False. True for cumulative distribution function, and false if not, which will be probability mass function.
Mean = 0.25 per 100000 Users. For a Population of 800000, mean will be = 8*0.25, Which is 1. Mean = 1.
a) What is the probability that a technician is required after a one-week period - If the technician is required than sale should have been > 3. Minimum 4. So lets find what is the probability for a sale = 4. Whcih is p(x=4). 1.5 % is the probability that a sale of 4 communication systems sale happens in a week This is Probablity Mass function.
But sale > 3 could be any number more than 3. Like P(X= 4,5,6....Infinite), which is not possible to calculate probability for each success and till infinitiy. So calculate probability for p(x<3), and negate it with Sample Space 1. Which will be 0.019. Hence its 1.9% probability that, a technician will be sent after a weeks time of your order. 1.9% is the Probablity of order placement more than 3 orders in a week. This is cumulative distribution Fucntion, probablity of sale happening any number more than 3.
b)What is the probability that no less than six orders have been placed in a two-week period? p(x > 6) = 1 - (p(x)<6)) = 0.008% (If mean is 1, considering week does not influence the mean, probability will be 45%, if mean is 2.
c. If you are the first in the city to place an order, what is the probability that a technician will not be dispatched for that week> = The sale has to be less than 3. p(x<3) = 98.10%