Probability

Function of one r.v. $Y = g(X)$ X 是离散，Y也是离散 X是连续，Y是离散 将Y的取值一一列出，再将Y的各种取值的概率求出 X是连续，Y是连续 $g(x)$是严格单调函...

正态（高斯）分布 $$ X \sim \mathcal{N}(\mu,,\sigma^{2}) $$ $$ p(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}, -\infty < x < \infty $$ $$ F(x) = \frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{x}e^{-\frac{(t-\mu)^2}{2\sigma^2}}dt $$ 如果RV $X \sim \mathcal{N}(\mu,,\sigma^{2})$, 则可以化为标准正态分布$U = \frac{X-\mu}{\sigma} \sim \mathcal{N}(0, 1)$ $E[X] = \mu$ $Var[X] = \sigma^2$ 若$X\sim \mathcal{N}(\mu, \sigma^2)$, $Y=aX+b \sim...

期望 RV的函数的期望是函数值和prob的积的积分或导数 $E[c] = c$, c是常数 方差 方差的定义：偏差平方的期望 偏差平方：$(X-E[X])^2$ $Var[X] = E[(X-E[X])^2]$ 性...

Markov inequality if X > 0 and E[X] is small, then X is unlikely to be very large markov 给出了RV大于某个值的概率的上界 if $X \geq 0$ and $a > 0$, then $P(X>a) \leq \frac{E[X]}{a}$ Chebyshev inequality if the variance is small, then X is unlikely to be too far from the mean chebyshev 不等式给...