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Exercise 5.3.5

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For a nonnegative variable $X$ , we have $Y = 0 \leq X$ when $X \leq a$ , and $Y = aX$ when $Xa$ . So in all cases $Y \leq X$ .

Then we have $E [Y] = 0 \times P (Xa) + aP (X \geq a) = aP (X \geq a)$ , also $E [Y] \leq E [X]$ , so we have $aP (X \geq a) \leq E [x]$ , so we have $P (X \geq a) \leq \frac{E [X]}{a}$ .

niuers

2020-03-20 00:00

Exercise 5.3.5

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