Continuous Random Variables

Cumulative Distribution Function

累计分布函数 cdf $F(x)$ 表示连续型随机变量 $X$ 不超过值 $x$ 的概率，即

$$F(x)=\mathbb{P}(X\le x)$$

由此可以延伸出 Probability Density Function pdf $f(x)$，其定义为

$$f(x)=\frac{d}{dx} F(x)$$

Support of $X$: we call area that $f(x)\gt 0$ as $\mathscr{S}$. Then we can rewrite the definition cdf as

$$\mathbb{P}(a\le X\le b)=\int_a^b f(x) dx$$

$$F(x)=\int_{-\infin}^x f(z) dz$$

Expectations of Continuous RV

Expectation (Mean) of Continuous RV is defined as

$$\mu_X=\mathbb{E}[X] = \int_{\mathscr{S}} x f(x) dx$$

And in general,

$$\mathbb{E}[g(X)] = \int_{\mathscr{S}} g(x) f(x) dx$$

Variance of Cont RV

The variance of $X$ is defined as

$$\sigma_X^2=\mathbb{E}[(X-\mu_X)^2]=\mathbb{E}[X^2]-\mu_X^2$$