Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Expected value ▷ Non-negativity

Theorem: If a random variable is strictly non-negative, its expected value is also non-negative, i.e.

\[\label{eq:mean-nonneg} \mathrm{E}(X) \geq 0, \quad \text{if} \quad X \geq 0 \; .\]

Proof:

1) If $X \geq 0$ is a discrete random variable, then, because the probability mass function is always non-negative, all the addends in

\[\label{eq:mean-disc} \mathrm{E}(X) = \sum_{x \in \mathcal{X}} x \cdot f_X(x)\]

are non-negative, thus the entire sum must be non-negative.


2) If $X \geq 0$ is a continuous random variable, then, because the probability density function is always non-negative, the integrand in

\[\label{eq:mean-cont} \mathrm{E}(X) = \int_{\mathcal{X}} x \cdot f_X(x) \, \mathrm{d}x\]

is strictly non-negative, thus the term on the right-hand side is a Lebesgue integral, so that the result on the left-hand side must be non-negative.

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Metadata: ID: P52 | shortcut: mean-nonneg | author: JoramSoch | date: 2020-02-13, 20:14.