Recovering Range Parameters
Octave, 82 bytes

function y=f(x)
while any(floor(y=linspace(x(1)+rand,x(end)+rand,numel(x)))-x),end

Running time is non-deterministic, but the code ends in finite time with probability 1.
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Explanation
The code defines a function of x that outputs y. The function consists of a while loop.
In each iteration, the right amount (numel(x)) of linearly spaced values are generated (linspace), starting at x(1)+rand and ending at x(end)+rand. These two calls to the rand function give random offsets between 0 and 1, which are applied to the initial and final values of x.
The loop is repeated for as long as any of the floored results differs (-)  from the corresponding entry in x.