The continuous random variable with uniform point inequality measure I(p)

By using the condition that the expected value of an absolute continuous random variable X is finite and positive and that the point inequality measure I(p) is uniform for 0<p<1, this paper discusses the question of the existence of such variable and proves that this problem has a unique solution.The obtained cumulative distribution function of X is a truncated Pareto distribution, with traditional inequality parameter equal to 0.5 and with support depending on the finite and positive mean and the level of uniformity of the point inequality measure I(p).