"Bridging" uniform and "mass" distributions

#1
My goal is to find a family X(a, b) of random variables (continuously) depending on two non-negative parameters a and b . The family should have the following properties:

(1) X(a, b) take values in the unit interval [0, 1] for all a, b and are not 1 almost surely;

(2) For dependent random variables Y(a, b) defined as 1/(a+b*X(a, b)) the expected values E[Y(a, b)] exist;

(3) When a tends to 0, E[Y(a, b)] tends to 1/b;

(4) When b tends to 0, E[Y(a, b)] tends to 1/a.

With regard to condition (3), note that E[Y(a, b)] = 1/(a+b) if X(a, b) has a ”mass” distribution: it equals 1 with probability 1 (and in fact does not depend on a and b). And if a tends to 0, then E[Y(a, b)] tends to 1/b as required in (3).

Also note that condition (4) is satisfied if all X(a, b) are uniformly distributed on [0, 1] (and in fact do not depend on a and b). Indeed, in this case we can calculate E[Y(a, b)] by integration to get that it equals (Ln(1 + b/a))/b. Since for small x Ln(1+x) is close to x, E[Y(a, b)] will be close to 1/a when b is close to 0.

So my goal is to find a family of random variables parameterized by a and b (may be one parameter b/a will suffice) to “bridge” the uniform and “mass” distributions.

I tried different parameterizations but was not able to find a parameterization satisfying all conditions (1)-(4).

I would appreciate any help or advice. Thank you.
 
Last edited:

Dason

Ambassador to the humans
#2
I tried different parameterizations but was not able to find a parameterization satisfying all conditions (1)-(4).

I would appreciate any help or advice. Thank you.
What things did you try? It'd be nice to know what doesn't work if I'm going to think about this some more.
 
#3
Thank you for your interest, Dason. I tried piecewise constant probability density functions and pdf’s that behave like x^(-p) for positive p when x is close to 0.