Fundamentals of Probability and Statistics for Engineers

178 Fundamentals of Probability and Statistics for EngineersTable 6.1 Observed frequencies (number ofobservations) of 0, 1, 2, . . . vehicles arriving in a30-second interval (for Example 6.11)No. of vehicles per 30sFrequency0 181 322 283 204 135 76 07 18 19 0Total 120a random manner both in space (amplitude and velocity) and in time (arrivalrate). Considering the time aspect alone, observations are made at 30-secondintervals as shown in Table 6.1.Suppose that the rate of 10 vehicles per minute is the level of critical trafficload. Determine the probability that this critical level is reached or exceeded.Let X(0,t) be the number of vehicles per minute passing some point on thepavement. It can be assumed that all conditions for a Poisson distribution aresatisfied in this case. The pmf of X(0, t) is thus given by Equation (6.44). Fromthe data, the average number of vehicles per 30 seconds is0…18†‡1…32†‡2…28†‡‡9…0†1202:08:Hence, an estimate of t is 2.08(2) ˆ 4:16. The desired probability is, then,P‰X…0; t† 10Š ˆX1p k …0; t† ˆ1ˆ 1X 9kˆ10kˆ0X 9kˆ0…4:16† k e 4:16k!1 0:992 ˆ 0:008:p k …0; t†The calculations involved in Example 6.11 are tedious. Because of its wideapplicability, the Poisson distribution for different values of t is tabulatedin the literature. Table A.2 in Appendix A gives its mass function for valuesof t ranging from 0.1 to 10. Figure 6.4 is also convenient for determiningTLFeBOOK