Let X be a random variable with E [ X ] = 1 and such...

tripes3h

tripes3h

Answered

2022-07-05

Let X be a random variable with E [ X ] = 1 and such that X 1 { X 0 } = 0 a.s.
Question Does it follow that X > 0 a.s.?
My progress I tried to write X = X + X , where both X + , X are nonnegative, but I did not make any progress...

Answer & Explanation

Valeria Wolfe

Valeria Wolfe

Expert

2022-07-06Added 11 answers

Observe that:
X 1 X 0 0 X 0 X 0 X < 0
so that
X 1 X 0 = 0 X 0
hence:
X 1 X 0 = 0  a.s. X 0  a.s.

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