Question

A study of four blocks containing 52 one-hour parking spaces was carried out and the results are given in the following table. Number of vacant one-hour parking spaces per observation period Observed frequency 2 3 26 31 45 20 15 7 3 Assuming that the data follow a Poisson distribution, determine: a) the mean number of vacant parking spaces, b) the standard deviation both ( i) from the given data and (ii) from the theoreti- cal distribution, and c) the probability of finding one or more vacant one-hour parking spaces, calculating from the theoretical distribution.

Answer 1

a)

the mean number of vacant parking spaces:

$\lambda=\sum x_1/n=\sum x_1/9=16.89$

b)

the standard deviation from the given data:

$\sigma=\sum (x_i-\lambda)^2/(n-1)=14.96$

the standard deviation from the Poisson distribution:

$\sigma=\sqrt{\lambda}=\sqrt{16.89}=4.11$

c)

for Poisson distribution:

$P(x=k)=\frac{\lambda^k e^{-\lambda}}{k!}$

the probability of finding one or more vacant one-hour parking spaces:

$P(x\ge 1)=1-P(0)=1-e^{-\lambda}=1-e^{-16.89}=0.999999954$