Normal Random Variable
- \( X \sim N(\mu, \sigma^2) \Rightarrow \frac{X - \mu}{\sigma} = Z \sim N(0,1) \).
這是一個非常OP的定理。因為它使得任何一般常態隨機變數的機率計算都可以轉化為標準常態隨機變數的機率計算。具體來說 - For any 0<a<b,
$$P(a<X<b)=\Phi( \frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})$$
where \( \Phi(x)=P(Z \leq x),\) cdf of \( Z \sim N(0,1).\) - Properties of \( \Phi, \phi\) (respectively, cdf and pdf of standard normal Z)
- \( X \sim N(\mu, \sigma^2)\)
- Verify that pdf of X is indeed a pdf
- \( E(X)=\mu,\ Var(X)= \sigma^2. \)
- Moment-generating function of X
$$M(t)=E(e^{tX}) = \exp({\mu t + \frac{\sigma^2 t^2}{2}}) $$
Moment-generating function
$$M(t)=E(e^{tX}). $$
- 小用：計算/生成moments: for k, a positive integer
$$E(X^k) = M^{(k)}(0)= M^{(k)}(t) |_{t=0} $$ if exists and \( M^{(k)}(t)\) is the k-th derivative of M(t). - 大用：Identify the random variable. If exists, moment-generating function uniquely determines a random variable. 簡單說，mgf 唯一決定一個隨機變數。也就是說，若有兩個隨機變數有相同的 mgf 則他們有相同的分配。