Foreword. The original formulation of this problem was inaccurate; chamomille and Didier Piau came up with a simple example which would not solve the problem in its accurate formulation. Sorry for my inaccuracy. Below is an edited version.
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;
(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 b/a is close to 0, the distribution of X(a, b) is close to uniform on [0, 1];
(4) When a/b is close 0, the distribution of X(a, b) is close to “mass“ distribution (that is, X(a, b) equals 1 with probability 1).
So my goal is to find a family of random variables parameterized by a and b to “bridge” the uniform and “mass” distributions.
I tried different parameterizations but was not able to find a parameterization satisfying all conditions (1)-(4).