# Thread: "Bridging" uniform and "mass" distributions

1. ## "Bridging" uniform and "mass" distributions

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.

2. ## Re: "Bridging" uniform and "mass" distributions

Originally Posted by 1stone
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. ## Re: "Bridging" uniform and "mass" distributions

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.

 Tweet

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts