Simulation from the Model

The increasing pool from which the Olympic competitors are chosen and a normal population with µ = 458 sec and σ = 50 sec might explain the decrease in winning times in the 1500 metres race.

The diagram below simulates this model — the 1896 winning time is the minimum of 20,000 values from the normal distribution and the subsequent winning times are minima from random samples of increasing size, up to a random sample of 2,000,000 values from the distribution in the year 2000.

Warning: It may take a considerable time for CAST to display the diagram below. Have patience!

Click Take sample again to see another sample from this model. (You may need a lot of patience if your computer is slow — the computer is generating millions or random numbers!)

Speeding up the Simulation

It takes a considerable time to generate each run of the simulation For example, in the year 2000, the winning speed is the minimum of 2 million random values which we have generated individually. Can we avoid generating so many random values?

The minimum of a random sample of n values also has a distribution. If we can obtain its distribution, we can directly generate the minimum by generating a single random value from this distribution, without generating the other n-1 values. Since only a single random value needs to be generated in each year, the simulations can be performed much more quickly. (A bit more about this faster algorithm is given later.)

The theory underlying this is a bit more advanced, but if you are interested, click the button on the right to read more detail about how these minima are generated.

Click Faster Algorithm in the diagram above, then repeat the simulation a few more times.

Applying a bit of probability theory may speed up your simulations considerably!