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For a real-valued random variable, $X$, the first moment method, is simply

$$P(X\ge\mathbb{E}[X])>0.$$

$\DeclareMathOperator\Var{Var}$This can be extended to the second moment quite easily (revised from original statement):

$$P(\lvert X-\mathbb{E}[X]\rvert\ge\sqrt{\Var[X]})>0.$$

The question must be asked: How does one generalize this to higher (probably centralized) moments?

Edit: Good catch Mark! Let me rephrase the question in another way.

Let $X$ be a real-valued random variable. Given only the first $n$ moments of $X$: $\mathbb{E}(X), \dotsc, \mathbb{E}(X^n)$, what is the largest value for $\lvert X-\mathbb{E}[X]\rvert$ that can be guaranteed to have positive probability?

For a real-valued random variable, $X$, the first moment method, is simply

$$P(X\ge\mathbb{E}[X])>0.$$

$\DeclareMathOperator\Var{Var}$This can be extended to the second moment quite easily:

$$\require{enclose}\enclose{horizontalstrike}{P(X\ge\mathbb{E}[X]+\sqrt{\Var[X]})>0}$$

$$P(\lvert X-\mathbb{E}[X]\rvert\ge\sqrt{\Var[X]})>0.$$

The question must be asked: How does one generalize this to higher (probably centralized) moments?

Edit: Good catch Mark! Let me rephrase the question in another way.

Let $X$ be a real-valued random variable. Given only the first $n$ moments of $X$: $\mathbb{E}(X), \dotsc, \mathbb{E}(X^n)$, what is the largest value for $\lvert X-\mathbb{E}[X]\rvert$ that can be guaranteed to have positive probability?

For a real-valued random variable, $X$, the first moment method, is simply

$P(X\ge\mathbb{E}[X])>0$

This can be extended to the second moment quite easily:

$P(X\ge\mathbb{E}[X]+\sqrt{Var[X]})>0$

$P(|X-\mathbb{E}[X]|\ge\sqrt{Var[X]})>0$

The question must be asked: How does one generalize this to higher (probably centralized) moments?

Edit: Good catch Mark! Let me rephrase the question in another way

Let $X$ be a real-valued random variable. Given only the first $n$ moments of $X$: $\mathbb{E}(X), \ldots, \mathbb{E}(X^n)$, what is the largest value for $|X-\mathbb{E}[X]|$ that can be guaranteed to have positive probability?

For a real-valued random variable, $X$, the first moment method, is simply

$$P(X\ge\mathbb{E}[X])>0.$$

$\DeclareMathOperator\Var{Var}$This can be extended to the second moment quite easily (revised from original statement):

$$P(\lvert X-\mathbb{E}[X]\rvert\ge\sqrt{\Var[X]})>0.$$

The question must be asked: How does one generalize this to higher (probably centralized) moments?

Edit: Good catch Mark! Let me rephrase the question in another way.

Let $X$ be a real-valued random variable. Given only the first $n$ moments of $X$: $\mathbb{E}(X), \dotsc, \mathbb{E}(X^n)$, what is the largest value for $\lvert X-\mathbb{E}[X]\rvert$ that can be guaranteed to have positive probability?

For a real-valued random variable, $X$, the first moment method, is simply

$P(X\ge\mathbb{E}[X])>0$

This can be extended to the second moment quite easily:

$P(X\ge\mathbb{E}[X]+\sqrt{Var[X]})>0$

The question must be asked: How does one generalize this to higher (probably centralized) moments?

For a real-valued random variable, $X$, the first moment method, is simply

$P(X\ge\mathbb{E}[X])>0$

This can be extended to the second moment quite easily:

$P(X\ge\mathbb{E}[X]+\sqrt{Var[X]})>0$

$P(|X-\mathbb{E}[X]|\ge\sqrt{Var[X]})>0$

The question must be asked: How does one generalize this to higher (probably centralized) moments?

Edit: Good catch Mark! Let me rephrase the question in another way

Let $X$ be a real-valued random variable. Given only the first $n$ moments of $X$: $\mathbb{E}(X), \ldots, \mathbb{E}(X^n)$, what is the largest value for $|X-\mathbb{E}[X]|$ that can be guaranteed to have positive probability?