# Random Numbers Biased

## Problem

Instead of a truly random number, you wish to randomly select a value from a set in which some values are more likely than others. For example, you may wish to simulate a normal distribution (i.e., a "bell curve") for a set of data.

## Solution

We will give a recipe for generating numbers with a normal distribution (aka Gaussian distribution, the bell shaped one). No existing library supports this functionality, though inspiration for one can be gained by viewing the Schemathics CVS. We will discuss a do-it-yourself method for explanatory purposes.

You will have to determine what kind of distribution you want, and locate the appropriate algorithm from a statistics reference.

For this recipe, we will consider the normal (Gaussian) distribution. If you need other distributions see either the CVS or consult a numerical analyst.

The function dis_var_new returns a stochastic variable (a thunk) with mean mu and standard deviance sigma.:

% derived from example in the documentation of SRFI27 % and translated to Erlang -record( dist_state, {state, mu, sigma} ). dist_var_new(Mu, Sigma) -> dist_var_new(false, Mu, Sigma). % create the thunk dist_var_new(State, Mu, Sigma) -> This = #dist_state { state = State, mu = 1.0 * Mu, sigma = 1.0 * Sigma}, IntPid = spawn(cookbook, dispatch, [This]), fun () -> IntPid ! {self(), value}, receive {retval, Any} -> Any end end. dispatch(This) -> receive {Pid, value} -> {NewThis, Value} = value(This), Pid!{retval, Value}, dispatch(NewThis) end. value(This) -> case This#dist_state.state of true -> Val = This#dist_state.mu + (This#dist_state.sigma * This#dist_state.state), {This#dist_state{state = false}, Val}; _ -> sigma_loop(This) end. sigma_loop(This) -> V1 = 2.0 * random:uniform() - 1.0, V2 = 2.0 * random:uniform() - 1.0, S = (V1 * V1) + (V2 * V2), if S >= 1.0 -> sigma_loop(This); true -> Scale = math:sqrt( (-2.0 * math:log(S)) / S), Val = This#dist_state.mu + (This#dist_state.sigma * Scale * V1), {This#dist_state{state = Scale * V2}, Val} end.

This is an interesting example, because it makes use of the fact that Erlang processes are so cheap to create and use. In effect, we've created a tiny server application who's only purpose is to listen for invocation requests and return a new normalized random number.

An example of usage:

1> X=dist_var_new(0, 1). #Fun<cookbook.4.38595032> 2> X1(). -0.873932 3> X1(). 4.91005e-2 4> X1(). 1.55993e-2 5> X1(). 0.181456

If you are unsatisfied with the fact that you get the same numbers as above, then randomize the source of the random numbers:

6> random:seed()

The algorithm used is the polar Box Muller method. The algorithm takes two independent uniformly distributed random numbers between 0 and 1 (present in the code as random:uniform()) and generates two numbers with a mean of my and standard deviation sigma. Note that the method produces two numbers at a time. Since we only need one, the second is saved for later in the variable next.

Note that the Perl Cookbook includes an interesting discussion of converting a set of values (and weights) into a distribution. This should also be converted to Erlang and shown here.

Mathematically-inclined Erlangers should also take a good look at Schemathics, which contains these and many other statistical methods.