A Characterization of the Exponential Distribution

This paper continues a study, initiated in [1], of the stochastic independence of $V = X - Y$ and the mean of order γ, {equation*}{1}M_γ{align*} = (1/γ) log (eγ X + eγ Y)/2 γ ≠ 0 and γ ≠ ± ∞\\ = (X + Y)/2 γ = 0\\ = max (X, Y) γ = + ∞\\ = min (X, Y) for γ = - ∞{align*}{equation*} for independent X and Y. The case γ = 0 is solved by the well known result, if, for independent X and Y, the variables $X + Y$ and $X - Y$ are independent, then both X and Y have normal distributions with a common variance. The first proof of this result may be found in Kac [6]. (Although the statement of this result there is much weaker, Kac's proof applies exactly to the statement above.) A corresponding result for the gamma distribution, proved in restricted form by Hogg [5], and proved without restrictions by Lukacs [7], is as follows: if Z₁ and Z₂ are independent positive random variables for which Z₁ + Z₂ and Z₁/Z₂ are independent, then both Z₁ and Z₂ have gamma distributions with a common scale parameter. This result may be used to complete the study of the independence of M_γ and V for independent X and Y when γ is finite and non-zero, by letting Z₁ = exp (γ X) and Z₂ = exp (γ Y). It is clear, then, from the result of Hogg and Lukacs that both exp (γ Y) and exp (γ Y) must have gamma distributions with a common scale parameter. The resulting distributions of X and Y have been studied in [1]. Therefore, of this study, there remain yet to be completely solved only the cases γ = +∞ and γ = - ∞. These two cases are essentially the same since if M+∞ and V are independent, then so are min (-X, -Y) and V. Thus, the negatives of the distributions for which M+∞ and V are independent will yield independent M-∞ and V. We consider only the case γ = -∞. This problem is considered in a previous paper, [2], in which the distributions were restricted to be discrete. The main result of that paper is as follows: if X and Y are independent, non-degenerate, discrete random variables, and if U = min (X, Y) and $V = X - Y$ are independent, then X and Y both have geometric distributions with common location and scale parameters, but possibly different geometric parameters. If the probability mass function of the geometric distribution is written as {equation*}{2}p(x) = (1 - p)p(x-θ)/c x = θ, θ + c, θ + 2c, ⋯{equation*} then, the parameter θ is a location parameter, c is a scale parameter, and p is called the geometric parameter, $c > 0, 0 < p < 1$. In this paper, the distributions of X and Y are restricted to be absolutely continuous. This will result in a characterization of the so-called exponential distribution whose density is {equation*}{3}f(x) = (1/σ) exp (-(x - θ)/σ) x > θ{equation*} and zero otherwise. The parameter θ is a location parameter, and the parameter σ > 0 is a scale parameter, which is also the standard deviation of the distribution. In these terms the main result of this paper may be stated as follows: if X and Y are independent random variables with absolutely continuous distributions, and if U = min (X, Y) and $V = X - Y$ are independent, then both X and Y have exponential distributions with a common location parameter but with possibly different scale parameters. This result then gives a characterization of the exponential distribution, since it is easy to show the converse, that if X and Y are independent and if each has an exponential distribution with a common location parameter but with possibly different scale parameters, then U and V are independent. For related results, the reader is referred to papers by Fisz [3] and Rogers [8].