Let X be a continuous variable for which the expectation E(X) exists. Show that the following holds:

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Let X be a continuous variable for which the expectation E(X) exists. Show that the following holds: image text in transcribed

(Hint: Making use of Proposition 6.7 for the nonnegative variable Y = |X|, express the expected value of Y as image text in transcribed

Then observe that for t ∈ [n, n + 1), we have P(Y ≥ n + 1) ≤ P(Y > t) ≤ P(Y ≥ n).)

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