Expectation of Normal Distribution

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Theorem

Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.

Then:

$\expect X = \mu$

where $\mu$ denotes the expectation of $X$.


Proof 1

From the definition of the normal distribution, $X$ has probability density function:

$\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$

From the definition of the expected value of a continuous random variable:

$\ds \expect X = \int_{-\infty}^\infty x \map {f_X} x \rd x$

So:

\(\ds \expect X\) \(=\) \(\ds \frac 1 {\sigma \sqrt {2 \pi} } \int_{-\infty}^\infty x \map \exp {-\frac {\paren {x - \mu}^2} {2 \sigma^2} } \rd x\)
\(\ds \) \(=\) \(\ds \frac {\sqrt 2 \sigma} {\sigma \sqrt {2 \pi} } \int_{-\infty}^\infty \paren {\sqrt 2 \sigma t + \mu} \map \exp {-t^2} \rd t\) substituting $t = \dfrac {x - \mu} {\sqrt 2 \sigma}$
\(\ds \) \(=\) \(\ds \frac 1 {\sqrt \pi} \paren {\sqrt 2 \sigma \int_{-\infty}^\infty t \map \exp {-t^2} \rd t + \mu \int_{-\infty}^\infty \map \exp {-t^2} \rd t}\)
\(\ds \) \(=\) \(\ds \frac 1 {\sqrt \pi} \paren {\sqrt 2 \sigma \intlimits {-\frac 1 2 \map \exp {-t^2} } {-\infty} \infty + \mu \sqrt \pi}\) Fundamental Theorem of Calculus, Gaussian Integral
\(\ds \) \(=\) \(\ds \frac {\mu \sqrt \pi} {\sqrt \pi}\) Exponential Tends to Zero and Infinity
\(\ds \) \(=\) \(\ds \mu\)

$\blacksquare$


Proof 2

By Moment Generating Function of Normal Distribution, the moment generating function of $X$ is given by:

$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

From Moment in terms of Moment Generating Function:

$\expect X = \map {M'_X} 0$

We have:

\(\ds \map {M_X'} t\) \(=\) \(\ds \frac \d {\d t} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}\)
\(\ds \) \(=\) \(\ds \map {\frac \d {\d t} } {\mu t + \frac 1 2 \sigma^2 t^2} \frac \d {\map \d {\mu t + \dfrac 1 2 \sigma^2 t^2} } \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \paren {\mu + \sigma^2 t} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}\) Derivative of Power, Derivative of Exponential Function

Setting $t = 0$:

\(\ds \map {M_X'} 0\) \(=\) \(\ds \paren {\mu + 0 \sigma^2} \map \exp {0 \mu + 0 \sigma^2}\)
\(\ds \) \(=\) \(\ds \mu \exp 0\)
\(\ds \) \(=\) \(\ds \mu\) Exponential of Zero

$\blacksquare$


Sources

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