# 概率密度函数

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{\displaystyle \forall -\infty <a<\infty ,\quad F_{X}(a)=\int _{-\infty }^{a}f_{X}(x)\,dx} \forall -\infty <a<\infty ,\quad F_{X}(a)=\int _{{-\infty }}^{{a}}f_{{X}}(x)\,dx

{\displaystyle \forall -\infty <x<\infty ,\quad f_{X}(x)\geq 0} \forall -\infty <x<\infty ,\quad f_{{X}}(x)\geq 0
{\displaystyle \int _{-\infty }^{\infty }f_{X}(x)\,dx=1} \int _{{-\infty }}^{{\infty }}f_{{X}}(x)\,dx=1
{\displaystyle \forall -\infty <a<b<\infty ,\quad \mathbb {P} \left[a<X\leq b\right]=F_{X}(b)-F_{X}(a)=\int _{a}^{b}f_{X}(x)\,dx} \forall -\infty <a<b<\infty ,\quad {\mathbb {P}}\left[a<X\leq b\right]=F_{X}(b)-F_{X}(a)=\int _{{a}}^{{b}}f_{{X}}(x)\,dx

{\displaystyle \mathbb {P} \left[X=a\right]=0} {\mathbb {P}}\left[X=a\right]=0，

{\displaystyle f_{\mathbf {I} _{[a,b]}}(x)={\frac {1}{b-a}}\mathbf {I} _{[a,b]}} f_{{{\mathbf {I}}_{{[a,b]}}}}(x)={\frac {1}{b-a}}{\mathbf {I}}_{{[a,b]}}

{\displaystyle f(x)={1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2} \over 2\sigma ^{2}}}} f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{- {{(x-\mu )^2 \over 2\sigma^2}}}

{\displaystyle \mathbb {E} [X^{n}]=\int _{-\infty }^{\infty }x^{n}f_{X}(x)\,dx} {\mathbb {E}}[X^{n}]=\int _{{-\infty }}^{{\infty }}x^{n}f_{X}(x)\,dx
X的方差为

{\displaystyle \sigma _{X}^{2}=\mathbb {E} \left[\left(X-\mathbb {E} [X]\right)^{2}\right]=\int _{-\infty }^{\infty }(x-E[X])^{2}f_{X}(x)\,dx} \sigma _{X}^{2}={\mathbb {E}}\left[\left(X-{\mathbb {E}}[X]\right)^{2}\right]=\int _{{-\infty }}^{{\infty }}(x-E[X])^{2}f_{X}(x)\,dx

{\displaystyle \mathbb {E} [g(X)]=\int _{-\infty }^{\infty }g(x)f_{X}(x)\,dx} {\mathbb {E}}[g(X)]=\int _{{-\infty }}^{{\infty }}g(x)f_{X}(x)\,dx[3]

{\displaystyle \Phi _{X}(j\omega )=\int _{-\infty }^{\infty }f(x)e^{j\omega x}\,dx} \Phi _{X}(j\omega )=\int _{{-\infty }}^{{\infty }}f(x)e^{{j\omega x}}\,dx