Characterizations motivated by the nexus between convolution and size biasing for exponential variables

Abstract

For a continuous density $f(x)$ with support on the real interval $(0,\infty )$ and finite mean $\mu $, its size biased density is defined to be of the form $(x/\mu )f(x).$ It is well known that for exponential variables, the convolution of two copies of the density yields the size biased form. This is the basis of the so-called inspection paradox. We verify that this agreement between size biasing and convolution actually characterizes the exponential distribution. We next consider the case in which the addition of one more term in a sum of independent identically distributed (i.i.d.) positive random variables also coincides with size biasing. Some related conjectures are also introduced. We then consider the problem of characterizing the class of all pairs of densities that can be called size-bias convolution pairs in the sense that their convolution is just a size biased version of one of them. We then consider discrete analogs to the size bias convolution results. It turns out that matters are more easily dealt with in the case of non-negative integer valued variables. Related geometric and Poisson characterizations are provided. Next, denote the sum of $n$ i.i.d non-negative integer valued random variables $\{X_i\}$, $i=1,2,...$ by $S_n$. We verify that the ratio of the densities of $S_{n_1}$ and $S_{n_2}$ determines the distribution of the $X$'s. The absolutely continuous version of this result, though judged to be plausible, can only be conjectured at this time.

Keywords

• Continuous density
• Size biased density
• Convolution
• Non negative integer valued variables
• Exponential distribution
• Size-bias convolution pairs.

References

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