Distributions
Suppose we simultaneously toss 500 coins and record the number that
come up heads. For several such experiments, the data can range from
zero to 500. If we plot number of times a particular result is
obtained (e.g., 300 were heads) versus that result, we get a
graph with tall bars for the most frequent results and short bars
for the infrequent results. If we toss only a few times, some
possibilities will not happen and there will be no bars for them. If
we have an enormous number of tosses, even the extremes of no heads
and all heads should have bars. As the number of tosses approaches
infinity, the bars will have a normal distribution. Even with no actual
data, the mathematics of normal distributions allows us to calculate
the odds for any of the possible results.
The estimates of distribution may be greatly in error when n, the number of observations, is small. The
distribution becomes better defined as n increases. We would like to
know the distribution when n approaches infinity. However, we reach
a point where adding more observations doesn't affect the
distribution noticeably.
Experiment with distributions
The applet shows "Number having this value" versus "Value or Number" for the random number generator of your computer. Use the Choice menu to get started, and try each of the cases. One thousand more random numbers are added each time that you click.
Things to try or consider:
Please decide whether this WHITE NOISE random number generator is reasonably good.
Why do you think that flipping coins is slow?
There is a Java routine that generates gaussian random numbers. Note that these give a normal distribution. Consider how you might change the shape of this normal distribution (higher and/or wider).
The option for flipping 500 coins tallies how many heads. It then plots Number having this value of heads versus the value, e.g., if no. of heads = 200, the total in the collector for 200 with heads is increased by one. Note that this is slow compared to the first option where a random number had to be generated once for each value. Now there must be 500 random numbers (flipping of the coin) to get each new value. Please decide whether this kluge for generating numbers has value. How would you change the width and/or height of the distribution?
The generator for gaussian numbers comes from the Java library. It is fast. My Java code has a bug for an extreme value, but the only symptom is no plot very infrequently.
Go to:
Normal Distributions
Pearson3 Distribution