Uniform distribution

We have already met the uniform distribution in Section 7.1, where we specified its PDF and CDF. To say that a random variable X is uniformly distributed on the interval [a, b], the notation X ∼ U(a, b) is used. We have already shown that the expected value is the midpoint on the support:

Since the uniform distribution is symmetric, the median and the expected value are the same, and skewness is zero. It can be shown that variance is

A peculiarity of the uniform distribution is that it has no well defined mode, since the PDF is constant. All of the remaining theoretical distributions have a single mode.

Fig. 7.11 PDF of a triangular distribution.

It is reasonable to say that the uniform distribution is a very dry model of uncertainty, as it just provides us with bounds on the possible realizations of X. It is often stated that the uniform distribution should be used whenever we have no idea about the underlying uncertainty. Actually this is a bit debatable, and the following argument has been proposed to counter this view. Suppose that the only thing we know about variable X is that it can take values between 0 and 1. Apparently, a uniform distribution U(0, 1) is an obvious choice. But now consider the variable Y = X^α, for some value α > 0. We cannot say anything about Y, either, and the variable is bounded between 0 and 1. However, we cannot say that both X and Y are uniformly distributed. Indeed, representing almost complete ignorance is not as easy as it may seem. Nevertheless, a uniform distribution is often used in Bayesian statistics as a noninformative prior.¹⁰ Another quite relevant application of U(0, 1) distribution is random-number generation for Monte Carlo simulation.¹¹ When we have to simulate randomness by a computer program, we first generate a U(0, 1) variable, which is then transformed to whatever we need to model uncertainty.