Question

Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter μ = 20 (suggested in the article "Dynamic Ride Sharing: Theory and Practice"†). (Round your answer to three decimal places.)

(a)

What is the probability that the number of drivers will be at most 13?

(b)

What is the probability that the number of drivers will exceed 26?

(c)

What is the probability that the number of drivers will be between 13 and 26, inclusive?

What is the probability that the number of drivers will be strictly between 13 and 26?

(d)

What is the probability that the number of drivers will be within 2 standard deviations of the mean value?

Answer 1

Poisson distribution is characterised by extremely low success probability... say p=0.05.That means there is large population of drivers commuting out of which the probability of particular occurance is 0.05.. Now inorder to find how large the population is , we divide the mean by its probabilty of occurance. ie 20/0.05=400. so since poisson distribution is assumed. with mean 20. we assume a large sample.. and the below probabilities cannot be computed directly using poisson formula as the computation will be tedious since it consumes a lot of time, the assumption of large sample is enough to compute the probabilities with the help of normal assumption.

In the case of poisson distribution we know that mean and variance are equal so Standard deviation = Square root of the variance=

a) Let X the random variable denoting the number of drivers between a particular origin and destination

The

  ( Standardisation of normal distribution Z is the standard normal variable)   5.94% of times that we see drivers less than 13

Similarly we can find the rest of the probabilities

b)P(X>26) (Give a try)

c)

=.9102-.0594 =0.8508

If some special calculators are used you can directly calculate the probabilities from poisson distribution but only slight difference can be observed in calculating the inclusive and not inclusive probabiilities

d)With in two standard deviation of the mean value actually means the probability of a standard normal variable actually lying between -2 and 2