Question

An Engineer in the Traffic Unit Department of a major urban centre was tasked to analyze a crash record for 10 intersections (with similar characteristics/circulating traffic volume) and answer the following questions: a) What is the season with the highest expected average crash frequeney? Estimate crash b) What is the probability of having more than 3 crashes/month for the season with the e) What is the month with lowest probability of having no crashes at intersections? And frequency as crashes/month. highest expected crash frequency? what is this probability? Table 1: Crash Record for 10 Intersections at Major Urban Centre 5 3 4 2 0 2 0 0 0 10 Assume crashes as Poisson distributed and use: SPRING March to May SUMMER-June to August FALL -September to November WINTER December to February

Answer 1

Seasonal Crashes per month for 10 intersections

Intersection	Spring	Summer	Fall	Winter
1	1	4	1.67	0.33
2	0.67	0.33	0	2
3	0	0.33	1	0.33
4	1	0.67	0.33	0
5	1	0	2.33	0.67
6	4.67	3	5	5.67
7	4.67	4.67	3.33	4.33
8	2	1	3	2.33
9	4.67	4.33	5	4
10	1.33	2	2.33	3.33
Average Crash frequency	2.1	2.03	2.4	2.3

The highest Average Crash frequency is in Fall season.

(b)

Let X ~ Poisson($\lambda$ = 2.4) where X is the number of crashes / month in Fall season.

P(X > 3) = 1 - P(X $\le$ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]

= 1 - [ exp(-2.4) * 2.4⁰ / 0! + exp(-2.4) * 2.4¹ / 1! + exp(-2.4) * 2.4² ​​​​​​​/ 2! + exp(-2.4) * 2.4³ ​​​​​​​/ 3! ]

= 0.2213

(c)

The month with the lowest probability of having no crashes at intersections is the month with the highest average crash frequency.

The average crash frequency for all 12 months are

2.5, 2, 2.3, 1.9, 2.1, 2.5, 2.0, 1.6, 2.7, 2.2, 2.3, 2.4

The month with the lowest probability of having no crashes at intersections is September month with average crash frequency = 2.7

Probability of having no crashes at intersections = exp(-2.7) * 2.7⁰ / 0! = 0.06720551