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For the random variable with probability density function \[ p(x) = \frac{1}{\sqrt{2\pi}\sigma}\exp \left\{ -\frac{(x-\mu)^2}{2\sigma^2} \right\} \] where \(\mu, \sigma > 0\) are parameters, we define \(X\) complies the normal distribution \(X\sim N(\mu, \sigma^2)\).
Regularity Checking.
Let \(\displaystyle x' = \frac{x - \mu}{\sigma}\), \[ F(+\infty) = \int_{-\infty}^{\infty} p(x)\dx = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}\sigma}\exp \left\{ -\frac{(x-\mu)^2}{2\sigma^2} \right\}\dx = \int_{-\infty}^{\infty} \varphi(x) = \frac{1}{\sqrt{2\pi}}\e^{-\frac{x'^2}{2}} \] Let \(x' = \sqrt{2t}\), \[ F(+\infty) = \]
Specially, when \(\mu=0, \sigma=1\), the normal distribution is called standard normal distribution, denoted as \(X\sim N(0, 1)\). The probability density function of standard normal distribution is denoted as \[ \varphi(x) = \frac{1}{\sqrt{2\pi}}\e^{-\frac{x^2}{2}} \] and its cumulative distribution function denoted as \(\Phi(x)\). Notice that \(\varphi(x)\) is even function hence \(\Phi(x) + \Phi(-x) = 1\).
Any normal distribution \(X\sim(\mu, \sigma^2)\) can be converted to the standard form by linear transformation: \[ Y = \frac{X-\mu}{\sigma} \sim N(0, 1) \]
Proof. The cumulative distribution function of constructed random variable \(Y\) is \[ F_Y(y) = P \left( \frac{X-\mu}{\sigma} \le y \right) = P(X\le \sigma y + \mu) = \int_0^{\sigma y + \mu} \frac{1}{\sqrt{2\pi}\sigma}\exp\left\{ -\frac{(x-\mu)^2}{2\sigma^2} \right\} \] Let \(x = \sigma y + \mu\), we have \[ F_Y(y) = \int_0^{y} \frac{1}{\sqrt{2\pi}\sigma}\e^{-\frac{y^2}{2}} \] and the probability density function \[ p_Y(y) = \frac{\d{F_Y(y)}}{\d y} = \frac{\d\left( \displaystyle \int_0^{y} \frac{1}{\sqrt{2\pi}\sigma}\e^{-\frac{y^2}{2}} \right)}{\d y} = \frac{1}{\sqrt{2\pi}\sigma}\e^{-\frac{y^2}{2}} \sim N(0,1) \]
The most simple form of normal distribution is
The uniform distribution is random variable with probability density function \[ p(x) = \left\{\begin{array}{ll} \cfrac{1}{b-a} & a\le x \le b\\ 0, & x<a, x>b \end{array}\right. \] denoted as \(X\sim U(a, b)\).