Given a sequence of random variables with Poisson$(\lambda)$, can we get a upper bound of $E[|X-E[X]|^k] $?
2026-04-01 03:48:32.1775015312
How to get the upper bound of $E[|X-E[X]|^k] $?
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$E[\vert X - E[X]\vert^k] = E[\vert X - \lambda\vert^k] = \sum_{i=0}^k (-1)^{i} {k \choose i} \lambda^{k-i}E[ X^i] \leq \lambda^k \exp(-2\lambda)\sum_{i=0}^k (-1)^{i} {k \choose i} \exp(i^2)$.
Now you could try to find simplify or bound the last term.