# Computing the Expectation of the Square of a Random Variable: $\text{E}[X^{2}]$.

What is the rule for computing $\text{E}[X^{2}]$, where $\text{E}$ is the expectation operator and $X$ is a random variable?

Let $S$ be a sample space, and let $p(x)$ denote the probability mass function of $X$.

Is $$\text{E}[X^{2}] = \sum_{x \in S} x^{2} \cdot p(x),$$ or do I also need to square the $x$ appearing in $p(x)$?

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Yes, that's it (as long as $X$ is discrete). – David Mitra Feb 18 '13 at 0:47
Yes, this is called second moment. – Alex Feb 18 '13 at 0:48
– Byron Schmuland Feb 18 '13 at 2:14
Dear CodeKingPlusPlus, you have to be a little careful with the formula that you’ve written down. The variable $x$ should only take values in the range of the random variable $X$. Just because $X$ is discrete doesn’t mean that the sample space $\Omega$ is finite or countably infinite. :) – Haskell Curry Feb 18 '13 at 7:01
Apart from that, the formula is correct! There’s no need to square the $x$ in the expression $p(x)$. – Haskell Curry Feb 18 '13 at 7:04

In general, if $(\Omega,\Sigma,P)$ is a probability space and $X: (\Omega,\Sigma) \to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ is a real-valued random variable, then $$\text{E}[X^{2}] = \int_{\Omega} X^{2} ~ d{P}.$$ Although this formula works for all cases, it is rarely used, especially when $X$ is known to have certain nice properties.
• If $X$ is a discrete random variable (i.e., its cumulative distribution function (cdf) is a step-function) and $p$ is its probability mass function (pmf), then we can use the formula $$\text{E}[X^{2}] = \sum_{x \in \text{Range}(X)} x^{2} \cdot p(x).$$
• If $X$ is an absolutely continuous random variable (i.e., its cdf is an absolutely continuous function), then it possesses a probability density function (pdf) $f$. We thus have the formula $$\text{E}[X^{2}] = \int_{\mathbb{R}} x^{2} f(x) ~ d{\mu(x)},$$ where $\mu$ is the standard Borel measure on $\mathbb{R}$. Of course, if $f$ is continuous, then we can simply compute the improper Riemann integral $$\text{E}[X^{2}] = \int_{- \infty}^{\infty} x^{2} f(x) ~ d{x}.$$