GATE Civil Engineering (CE) 2019 Shift 1 Solved Paper

Question : 41 of 65

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Traffic on a highway is moving at a rate 360 vehicles per hour at a location. If the number of vehicles arriving on this highway follows Poisson distribution, the probability (round off to 2 decimal places) that the headway between successive vehicles lies between 6 and 10 seconds is __________

Your Answer:

Validate

Solution:

λ

(rate)

=360

vehicle/hour

=0.1

vehicle/hour
Let '

X^{'}

denote the headway between two successive arrivals. [clearly exponential R.V].

∴ {f}( {n})=λ {e}^{-λ x}[

pdf of

{x}] →

exponential distribution

⇒ {f}( {n})=0.1 {e}^{-1.0 {x}}

∴

Required probability

{P}[6 < {x}↙{↓↙{6 < x <, 10}} < 10]=∫↙{6}↖{10} {f}( {n}) {dx}

=∫↙{6}↖{10} 0.1 {e}^{-01 {x}}=0.1 \;{[ {e}^{-0.1 {x}}]}/{(-0.1)}|_{6} ^{10}=-[ {e}^{-0.1(10)}- {e}^{-0.1 {/} 6)}]=-[ {e}^{-1}- {e}^{-0.6}]= {e}^{-06}- {e}^{-1}

⇒ {P}[6< {x}<10]=0.18

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