Normal Distribution: Mean and Variance
Formula
$X \sim N\left( \mu , \sigma^{2} \right)$ Plane $$ E(X) = \mu \\ \operatorname{Var} (X) = \sigma^{2} $$
Derivation
Strategy: The normal distribution has a moment-generating function that is easy to differentiate, so we just directly derive it.
Moment-generating function of normal distribution: $$ m(t) = \exp \left( \mu t + {{ \sigma^{2} t^{2} } \over { 2 }} \right) \qquad , t \in \mathbb{R} $$
$$ m ' (t) = \left( \mu + \sigma^{2} t \right) \exp \left( \mu t + {{ \sigma^{2} t^{2} } \over { 2 }} \right) $$ Therefore, it is $E(X) = m ' (0) = \mu$, and $$ m '' (t) = \left( 0 + \sigma^{2} \right) \exp \left( \mu t + {{ \sigma^{2} t^{2} } \over { 2 }} \right) + \left( \mu + \sigma^{2} t \right)^{2} \exp \left( \mu t + {{ \sigma^{2} t^{2} } \over { 2 }} \right) $$ Therefore, it is $E \left( X^{2} \right) = m '' (0) = \sigma^{2} + \mu^{2}$. Thus, it is $\operatorname{Var} (X) = \sigma^{2}$.
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