Proof: Non-negativity of the expected value
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Expected value ▷ Non-negativity
Metadata: ID: P52 | shortcut: mean-nonneg | author: JoramSoch | date: 2020-02-13, 20:14.
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
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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Sources: - Wikipedia (2020): "Expected value"; in: Wikipedia, the free encyclopedia, retrieved on 2020-02-13; URL: https://en.wikipedia.org/wiki/Expected_value#Basic_properties.
Metadata: ID: P52 | shortcut: mean-nonneg | author: JoramSoch | date: 2020-02-13, 20:14.