Question: Counterexample. Let X be a random variable taking on the values 1, 0, 1, 2, with probabilities P{X = 1} = ; P{X =

Counterexample. Let X be a random variable taking on the values

−1, 0, 1, 2, … with probabilities Pθ{X = −1} = θ; Pθ{X = x} = (1 − θ)

2

θx , x = 0, 1,....

Then P = {Pθ, 0 < θ < 1} is boundedly complete but not complete. [Girschick et al. (1946)]

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