This simulation shows the statistical nature of the radioactive decay; it also helps to understand certain statistical concepts (expectation value, scatter).
The program simulates an experiment performed using a Geiger-Müller counter. The built-in random number generator simulates the counts of the GM-counter. For easier observation the time can be accelerated by a factor of ten (setting the Fast checkboxl).
There are two modes of the program: Poisson-distribution, and Half life. These modes can be selected using the radio buttons at the right hand side of the screen. (The explanation below refers to the actual operation mode).
Poisson-distribution
In this operating mode only the constant intensity"Background" can be set. The reason for that is that the counts follow the Poisson-distribution only if the expectation value is constant. The counter measures the counts in every second. The counts and the time will be shown, and they are displayed in the graph in the right hand side of the screen.
In the Poisson-distribution operating mode we count how many times we got 0, 1, 2... etc. counts in 1 sec.
During a real measurement the "theoretical" value of the intensity (which can be set here) is not known, only the measured values are known. Using the measured values some parameters should be deduced that approximate the "theoretical" values, if the number of the measurements are sufficiently large.
Therefore the program calculates the expectation value and scatter using exclusively the measured values as one would calculate them at a real measurement. These values are shown in the upper right corner of the graph-window, and it "fits" a theoretical Poisson-distribution curve on the measured points (red curve).
It is interesting to note that the calculated values get closer and closer to the "theoretical" values (given in the program) as the number of experimental points increase. This helps to understand that although the individual events are stochastic, for large numbers the statistical laws govern the behaviour, wich can be well described by notions like expectation value, and scatter around it.
Please note that the expectation value, and scatter are parameters of the theoretical distribution. The measured quantities are the mean and the standard deviation. These get closer and closer to the theoretical values as the number of the measurement points get larger and larger.