5Probability Distributions
5.1 A Random Experiment
Imagine an archery contest. Figure 5.1 shows the scatter of hits on a target, as it might appear after 1000 arrows have been shot. We are all familiar with the fact that such patterns arise as a result of random events. It is not possible to predict with precision where the next shot will land, but one can predict with confidence that most hits will be in the neighbourhood of the aiming point (the bull's‐eye with coordinates (
) in the figure), while a proportion of bad shots will be further away. The scatter shown in the figure (which was simulated on a computer, needless to say) is what might be obtained if the average magnitude of the aiming errors was the same in all directions – undershooting as likely to happen as overshooting, and so forth. It has also been assumed (reasonably?) that the vertical and horizontal deviations of the hits are not correlated – shooting too high/low is not systematically connected with a bias to left/right.

Figure 5.1 Archery target scatter.
Randomness can be a difficult concept to make sense of. Archers fail to hit the mark they aim at for a variety of reasons, from wind variations to imperfect muscular control. One feels that such factors could in principle be understood and, with sufficient data, be ...