S2 Three Uniforms

Picture below shows densities of THREE different uniform variables. U[-3,3] has density of 1/6 going from -3 to 3. The base of the rectangle is 6 units, while the height is 1/6 so the area is 1. Similarly U[-1.5,+1.5] has density constant at 1/3 between -1,5 and 1.5; again the area of the rectangle is 1. Finally the uniform density between [-0.5,0,5] has constant 1, and the rectangle formed by the density has area 1.

What happens if we consider a uniform density on [-a,a] where a goes to 0? The height keeps increasing as the base keeps getting smaller, keeping the area constant at 1. In the limit, we get a DISCRETE random variable which equals 0 with probability 1. This is also called Point Mass at 1. This is, of course, a useful model of many types of real world situations.

What happens if we consider the uniform distribution [-N.N] where N goes to infinity? This leads to a random variable which is equally likely to take any value on the real line between plus or minus infinity. The base of the rectangle is infinite, so to make the area of the density one, the value of the density must be infinitesmal. This density is useful as a prior in some Bayesian calculations.