Perceived Highway Speed Jenny Switkes Cal Poly Pomona

Getting Started: A Discrete Example ½ of cars driving at 40 mph (“slow”) ½ of cars driving at 40 mph (“slow”) ¼ of cars driving at 60 mph (“medium”) ¼ of cars driving at 60 mph (“medium”) ¼ of cars driving at 80 mph (“fast”) ¼ of cars driving at 80 mph (“fast”) Average highway speed – a weighted average! Average highway speed – a weighted average!  expected value

½ at 40 mph; ¼ at 60 mph; ¼ at 80 mph Average perceived by “slow” drivers: Average perceived by “slow” drivers:

½ at 40 mph; ¼ at 60 mph; ¼ at 80 mph Average perceived by “medium” drivers: Average perceived by “medium” drivers:

½ at 40 mph; ¼ at 60 mph; ¼ at 80 mph Average perceived by “fast” drivers: Average perceived by “fast” drivers:

½ at 40 mph; ¼ at 60 mph; ¼ at 80 mph Weighted average of the perceptions: Weighted average of the perceptions: Actual average highway speed: Actual average highway speed:

A Necessary Detour towards Our Destination: Continuous Random Variables

An Intro to Continuous Random Variables

Some Famous Random Variables

Moving to the continuous model…

PDF for velocities of vehicles we “see”

True average: True average:

An Example

Some Questions… Under what conditions will our perception be accurate? Under what conditions will our perception be accurate? If our perception is inaccurate, is it too high or too low? If our perception is inaccurate, is it too high or too low? How can we measure the average speed accurately? How can we measure the average speed accurately?

Denominator is positive. Denominator is positive.

Focusing on the error’s numerator, letting w =  x;  =  Focusing on the error’s numerator, letting w =  x;  =  -s

Symmetric PDF – Conclusions w =  x;  =  -s w =  x;  =  -s If s= , then  =0 and error is zero. If s= , then  =0 and error is zero. Average driver’s perception is accurate! Average driver’s perception is accurate! If s> , then  , then  <0 and error is positive. Fast driver’s perception is low. If s 0 and error is negative. Slow driver’s perception is high. If s 0 and error is negative. Slow driver’s perception is high.

 A sufficient condition: If the distribution of vehicle speeds is symmetric about our speed, then our perception is accurate.  The condition given above is not necessary: We can find weird, non- symmetric distributions for which our perception can be accurate.

If our perception is inaccurate, is it too high or too low? If our perception is inaccurate, is it too high or too low? From our earlier work,  For symmetric distributions, an observer traveling slower than the true mean speed will perceive a mean speed higher than the true mean speed. The reverse happens for someone traveling faster than the true mean speed.

 If we are driving at the perceived median speed, our speed is the actual mean speed of the vehicles.

An Extension: What if we average the speed of vehicles as perceived by the overall collection of vehicles on the road?

Average speed of vehicles as perceived by the overall collection of vehicles on the road:

Symmetric PDF – Many Drivers After a little bit of calculus, we find: After a little bit of calculus, we find:

Symmetric PDF – Many Drivers

For a symmetric distribution of vehicle speeds, while slower vehicles overestimate the average highway speed, and faster drivers underestimate the average highway speed, these effects are symmetric. For a symmetric distribution of vehicle speeds, while slower vehicles overestimate the average highway speed, and faster drivers underestimate the average highway speed, these effects are symmetric. The average of the drivers’ perceptions gives the correct average highway speed  The average of the drivers’ perceptions gives the correct average highway speed 

A general condition for average of drivers’ perceptions to be accurate. If the integral If the integral then then Changed the order of integration!

What else could be investigated? Look in detail at specific distributions of vehicle speeds (gamma, normal). Look in detail at specific distributions of vehicle speeds (gamma, normal). Other ways to accurately perceive the speed of traffic on the highway. Other ways to accurately perceive the speed of traffic on the highway. Other conditions for the average of the drivers’ perceptions to be the true average. Other conditions for the average of the drivers’ perceptions to be the true average. Further ideas? Further ideas?

References Dawson, B. and Riggs, T. (2004). Highway relativity. College Mathematics Journal 35, Dawson, B. and Riggs, T. (2004). Highway relativity. College Mathematics Journal 35, Haight, F.A. (1963). Mathematical Theories of Traffic Flow. Academic Press, New York. Haight, F.A. (1963). Mathematical Theories of Traffic Flow. Academic Press, New York. Stein, W. and Dattero, R. (1985). Sampling bias and the inspection paradox. Mathematics Magazine 58, Stein, W. and Dattero, R. (1985). Sampling bias and the inspection paradox. Mathematics Magazine 58, Swift, R., Switkes, J. and Wirkus, S. (2003). Perceived highway speed. The Mathematical Scientist 28, Swift, R., Switkes, J. and Wirkus, S. (2003). Perceived highway speed. The Mathematical Scientist 28, Swift, R., Switkes, J. and Wirkus, S. (2006). On highway relativity. The Mathematical Scientist 31, Swift, R., Switkes, J. and Wirkus, S. (2006). On highway relativity. The Mathematical Scientist 31,

Thank you! Any other questions/thoughts/ideas? Any other questions/thoughts/ideas?