In the article on potholes I mentioned the word simulation. This is a technique frequently used to mirror real life situations. Basically it consists of selecting a random sample from a known distribution and using that sample for modelling, testing, etc. The selection of the sample is based upon an important idea, namely that the cumulative distribution function of any random variable is itself a random variable with a (0,1) uniform distribution. More about this in the next issue.
Suppose X is an exponential random variable with density function
f(x) = e-x, x > 0 and cumulative distribution function F(a) = P(X < a) = 1 - e-a, a > 0.
Suppose now we define Y = 1 - e-X, where X is as above. Prove, by
confirming each of the following statements, that Y is a random
variable with (0,1) uniform distribution:
(1) 0 £ Y £ 1 for all X > 0,
(2) F(b) = P(Y £ b) = b, 0 £ b £ 1, and (3) ò01 f(y) dy = 1.
Send your solutions to the editor by 15 Sep - prizes will be awarded for good solutions.