Random Numbers
thanhnga la (tla@ucla.edu)
Tue, 31 Oct 95 03:19:48 -0800
To answer the reader's question, we first consider the basic random
number from which almost all computer-based random generation springs is
the uniform, or iquidistributed random variable. Once this random
variable is in the hand, more general probability distributions can be
obtained numerically. In what follows we shall assume a probability
density p(x) and a probability distribution function P(x) such that:
p(x) >= 0; integral (from -infinity to +infinity of) p(x)dx = 1
P(x) = integral (from -infinity to x of) p(u)du
Thus, P is monotonic non-decreasing on (0,infinity). For example, an
equidistributed random variables is taken to have density function as the
characterized constant function on (0,1), so that in this case P is
linear ramp.
The most direct theoretical way to create a random variable with a
prescribed distribution is the inverse method. One creates numbers:
y = P^-1(x)
where x is taken to be equidistrubuted on (0,1). It is not hard to see
that the y have probability distribution P.
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