tripes3h

2022-07-05

Let X be a random variable with $E\left[X\right]=1$ and such that $X{1}_{\left\{X\le 0\right\}}=0$ 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...

Valeria Wolfe

Expert

Observe that:
$X{1}_{X\le 0}\ne 0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}X\ne 0\wedge X\le 0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}X<0$
so that
$X{1}_{X\le 0}=0\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}X\ge 0$
hence:

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