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I'm trying to answer the following question from Larry Wassermans book on statistical inference.

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My question is how did they arrive at the Markov bound, it does not seem like the normal form of the Markov inequality i.e. $P(Z > t) \leq \frac{\mathbb{E}(Z)}{t}$. Also how could we analytically calculate $\mathbb{E}(|Z|^k)$.

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  • $\begingroup$ Apply the "normal form" of the equality to the random variable $|Z|^k.$ $\endgroup$ – whuber May 21 '19 at 12:04
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First note that the Markov inequality applies to non-negative random variables; you can't apply it directly to $Z$.

Let $Y=|Z|^k$ for $k>0$, which is non-negative. Now apply the Markov inequality to $Y$, then relate that back to a probability statement about $|Z|$. (Edit: as whuber already noted in comments; not sure how I missed that)

As for evaluating $E(|Z|^k)$, there's several ways to approach it. Let's take a fairly straightforward one -- just write the integral, use the symmetry of $|Z|$ to write it in terms of an integral on the positive half line, and then make an obvious substitution to get a gamma function.

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