Let X be uniformly distributed on (, + 1), 0 v. [Basu (1964).]

Let X be uniformly distributed on (θ, θ + 1), 0 <θ< ∞, let

[X] denote the largest integer ≤ X, and let V = X − [X].

(i) The statistic V (X) is uniformly distributed on (0, 1) and is therefore ancillary.

(ii) The marginal distribution of [X] is given by

[X] =  [θ] with probability 1 − V (θ),

[θ] + 1 with probability V (θ).

(iii) Conditionally, given that V = v, [X] assigns probability 1 to the value [θ]

if V (θ) ≤ v and to the value [θ] + 1 if V (θ) > v. [Basu (1964).]