PROBABILITY METHOD IN TRAFFIC ENGINEERING

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PROBABILITY METHOD IN TRAFFIC ENGINEERING
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PROBABILITY METHOD

• Poisson Distribution
• Normal Distribution
• Binomial Distribution

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BINOMIAL DISTRIBUTION

• Consider an observation in which a particular
  result may or may not vehicle break a speed
  limit.
• For example, a number of vehicle is observation
  and it may be acceptable or not be acceptable
  speed limit

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BINOMIAL DISTRIBUTION

• EXAMPLE ( All Units in km/h)
• 63, 63, 55, 58, 68, 61, 68, 63, 55, 62, 72, 57,
  61, 68, 78, 64, 63, 51, 59, 75, 70, 74, 58, 69,
  65, 66, 58, 72, 74, 75, 60, 73, 69, 57, 63, 65,
  72, 59

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BINOMIAL DISTRIBUTION

• nCk pk q n-k
• p the probability is acceptable or occur
• q the probability is not acceptable or occur
• 1 p
• n number of observation
• k frequency

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BINOMIAL DISTRIBUTION

• Let say, speed limit that particular road is 60
  km/h. Probability vehicle not break speed limit,
  p 11 / 38
• 0.285
• The first 10 vehicles observe, only 2 vehicle
  not break speed limit
• So, n 10 and k 2

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BINOMIAL DISTRIBUTION

• P (vehicle not break speed limit)
• nCk pk q n-k
• 10C2 0.2852 0.715 10-2
• 45 ( 0.082) (0.0683)
  
• 0.252
• 25.2

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POISSON DISTRIBUTION

• Know as counting distribution. It has the clear
  physical meaning of a number of events X
  occurring in a specified counting interval of
  duration T
• Indeed, the Poisson would have been a suitable
  approximation to the low p, high n cases in the
  preceding subsection

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POISSON DISTRIBUTION

• P( X r ) e -? ?r / r!
• ? Mean of distribution
• Example Probability driver drive 54km/h
• from group data, ? 64.579 km/h,
• P( X 54 ) e -64.579 64.57954 / 54!
• 8.987x10-29 5.558x1097/2.308x1071
• 0.0216
• 2.16

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NORMAL DISTRIBUTION

• A variable which can take on any value within a
  given range is called a continuous variable.
• The normal distribution allows us to calculate
  the probability of continuous variable falling
  within a particular range of value
• The normal probability density function is a
  bell-shaped curve

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NORMAL DISTRIBUTION

• Z (x µ) / s
• µ mean
• s standard deviation

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NORMAL DISTRIBUTION
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NORMAL DISTRIBUTION

• Single operation
• P (x lt a) ? Direct Read From Table (DRT)
• P (x gt a) ? (1 DRT)
• P (x lt -a) ? ( 1 DRT)
• P (x gt -a) ? (DRT)

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NORMAL DISTRIBUTION

• For Example group data, µ 65 and s 7
• Z (x µ) / s
• P (x lt 54) P (Z lt (54 - 65)) / 7)
• P( Z lt -11 / 7)
• P ( Z lt -1.57)
• 1 0.9418
• 0.0582
• 5.82

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NORMAL DISTRIBUTION

• P (x gt 54) P (Z gt (54 - 65)) / 7)
• P( Z gt -11 / 7)
• P ( Z gt -1.57)
• 0.9418
• 94.18

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NORMAL DISTRIBUTION

• P (x lt 74) P (Z lt (74 - 65)) / 7)
• P( Z lt 9 / 7)
• P ( Z lt 1.29)
• 0.9015
• 90.15

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NORMAL DISTRIBUTION

• P (x gt 74) P (Z gt (74 - 65)) / 7)
• P( Z gt 9 / 7)
• P ( Z gt 1.29)
• 1- 0.9015
• 0.0985
• 9.85

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NORMAL DISTRIBUTION

• Double operation
• a lt x lt b ? P ( x lt b) P (x lt a)
• DRT - DRT
• -a lt x lt b ? P( x lt b) P (x lt -a)
• DRT - (1-DRT)
• -a lt x lt -b ? P ( x lt -b) P ( x lt -a)
• (1 DRT) (1 DRT)

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NORMAL DISTRIBUTION

• P(54 lt x lt 74) P( Zlt(74-65)/7) P(Zlt(54-65)/7)
• P ( Z lt 1.29) P (Z lt
  -1.57)
• 0.9015 - (1 - 0.9418 )
• 0.9015 - 0.0582
• 0.8433
• 84.33

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NORMAL DISTRIBUTION

• P( 70lt x lt74) P( Zlt(74-65)/7) P(Zlt(70-65)/7)
• P ( Z lt 1.29) P (Z lt
  0.71)
• 0.9015 - 0.7611
• 0.1404
• 14.04

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NORMAL DISTRIBUTION

• P(54lt xlt60) P( Zlt(60-65)/7) P(Zlt(54-65)/7)
• P ( Z lt -0.71) P (Z
  lt -1.57)
• (1 - 0.7611) - (1 -
  0.9418 )
• 0.2389 - 0.0582
• 0.1807
• 18.07