Question

An observer counts 240 veh/h at a specific highway location. Assume that the vehicle arrival at the location is Poisson distributed, the probability of having one vehicle arriving over a 30-second time interval is ____________

1. 0.27
2. 0.31
3. 0.29
4. 0.30

Answer 1

Correct Answer - Option 1 : 0.27

Concept:

Poisson Distribution

The Poisson distribution can be used to model the vehicle arrival

\({\rm{P}}\left( {{\rm{n}},{\rm{t}}} \right) = \frac{{{{\rm{e}}^{ - {\rm{λ t}}}}{{\left( {{\rm{λ t}}} \right)}^{\rm{n}}}}}{{{\rm{n}}!}}\)

Where,

λ = Number of vehicles per second

t = Time interval

n = Number of vehicles

Calculation:

Given,

λ = 240 veh/hr = 240/3600 veh/sec = 0.0666 veh/sec

n = 1, t = 30sec

\({\rm{P}}\left( {{\rm{n}},{\rm{t}}} \right) = \frac{{{{\rm{e}}^{ - {\rm{λ t}}}}{{\left( {{\rm{λ t}}} \right)}^{\rm{n}}}}}{{{\rm{n}}!}}\)

\({\rm{P}}\left( {{\rm{1}},{\rm{30}}} \right) = \frac{{{{\rm{e}}^{ - {\rm{0.0666× 30}}}}{{\left( {{\rm{0.0666× 30}}} \right)}^{\rm{1}}}}}{{{\rm{1}}!}}\)

P (1, 30) = e^-2 × 2 = 0.27