1. 1 Theoretical Physics Experimental Physics Equipment, Observation Gambling: Cards, Dice Fast PCs Random- number generators Monte- Carlo methods Experimental Statistics Theoretical Statistics Then Very recent Monte Carlo Simulation (MCS) This is a kind of “experimental statistics”. In other branches of science, for example physics, the relationship between theory and experiment can be depicted in this way: In statistics, theory developed from simple observations in card and dice games in the 17 th century and later. The fully-fledged “experimental” approach, now known as Monte Carlo and thus acknowledging the origins of statistics in gambling, had to await the development of fast personal computers and random-number generators:

2. 2 Example of use of Monte Carlo: Sample of n measurements: x x x x 1 2 3 n with mean x,,, …….. (x - x) + (x - x) + (x - x) + (x - x) 1 2 3 n 2 2 2 2 + …. (n – 1) Unbiased estimate of population variance is calculated as: The sum of squares is divided by (n – 1). If divided by n, the estimate would be biased too low. How can this be shown using Monte Carlo?

3. 3 For each sample of 4, calculate the mean and unbiased variance using the formula with the correct (n – 1) = (4 – 1) = 3. Calculate and print out the average of these 25 000 means and 25 000 variances. For each sample of 4, repeat the calculation dividing the sum of squares by 4 instead of 3. Calculate and print out the average of these 25 000 variances. Next slide shows results of an actual Monte Carlo. Use Gaussian (normal) random-number generator to generate (say) 100 000 numbers with mean 0 and standard deviation 1 (variance 1). Divide up these 100 000 numbers into 25 000 samples of 4 each (so n = sample size = 4).

4. 4 100 000 random numbers generated with Gaussian distribution and mean 0, standard deviation 1 (variance 1). Average of means of 25 000 samples of 4 each: 0.000184 Correct value: 0 Average of variances of 25 000 samples of 4 each (using correct divisor n – 1 = 4 – 1 = 3): 0.999975 Average of variances of 25 000 samples of 4 each (using incorrect divisor n = 4): 0.749979 Correct value: 1 so divisor n – 1 = 3 gives a much closer estimate of the population variance.

5. 5 100 000 random numbers generated with uniform distribution between 0 and 1. Mean: 0.5, variance 1/12 = 0.083333. Average of means of 25 000 samples of 4 each: 0.499897 Correct value: 0.5 Average of variances of 25 000 samples of 4 each (using correct divisor n – 1 = 4 – 1 = 3): 0.083860 Average of variances of 25 000 samples of 4 each (using incorrect divisor n = 4): 0.062895 Correct value: 1/12 = 0.083333. so divisor n – 1 = 3 gives a closer estimate of the population variance (but not as close as in the Gaussian case).