📂Probability Distribution

Normal Distribution: Mean and Variance

Formula

$X \sim N\left( \mu , \sigma^{2} \right)$ Plane $$E(X) = \mu \\ \Var (X) = \sigma^{2}$$

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: $$m(t) = \exp \left( \mu t + {{ \sigma^{2} t^{2} } \over { 2 }} \right) \qquad , t \in \mathbb{R}$$

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$$m ' (t) = \left( \mu + \sigma^{2} t \right) \exp \left( \mu t + {{ \sigma^{2} t^{2} } \over { 2 }} \right)$$ 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 $\Var (X) = \sigma^{2}$.

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