Normal Distribution
Definition 1. We say a random variable $X$ has a normal distribution if its pdf is $$ f(x) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left\{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2\right\}, \forall x \in \mathbb{R},$$ denoted by $X \sim N(\mu, \sigma^2)$.
Let $X \sim N(\mu, \sigma^2)$.
(i) Expectation: $\mu = E(X)$.
(ii) Variance: $\sigma^2 = Var(X)$
(iii) Mgf: $$M(t) = \exp \left( \mu t + \frac{1}{2} \sigma^2 t^2 \right), \forall t \in \mathbb{R}$$
De Moivre-Laplace Theorem은 이항분포를 따르는 확률변수의 표본 크기가 무한히 많아진다면 정규분포로 수렴함을 말해준다. Central limit theorem의 특수한 경우라고도 볼 수 있다. 감마 분포에서 $\alpha$와 $\beta$가 각각 pdf 곡선의 위치와 모양을 결정했듯이, 정규 분포에서 평균과 분산은 정규분포곡선의 위치와 스케일을 결정한다고 볼 수 있다.
Remark
Remark.
(i) $X \sim N(\mu, \sigma^2) \iff X \overset{D}{=} \sigma Z + \mu, \quad Z \sim N(0, 1)$
(ii) $X \sim N(\mu, \sigma^2) \iff aX + b \sim N(a\mu + b, a^2\sigma^2)$
(iii) $$X_i \overset{indep}{\sim} N(\mu_i, \sigma_i^2), \forall i = 1, ..., n \\ \Longrightarrow Y = \sum_{i=1}^n a_i X_i \sim N\left( \sum_{i=1}^n a_i \mu_i, \sum_{i=1}^n a_i^2 \sigma_i^2 \right)$$ (iv) $X_1, \ldots, X_n \overset{iid}{\sim} N(\mu, \sigma^2) \Longrightarrow \overline{X} \sim N\left( \mu, \frac{\sigma^2}{n} \right)$
Standard Normal Distribution
Definition 2. Let $X \sim N(\mu, \sigma^2)$. The random variable $Z = \frac{X - \mu}{\sigma}$ is said to have a standard normal distribution, denoted by $Z \sim N(0, 1)$. We often denote the cdf and pdf of $Z$ by $\Phi(z)$ and $\phi(z)$, respectively.