تابع چگالی احتمال

${\displaystyle Pr(a\le X\le b)={\int }_a^{b}f(x)dx}$

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1}$

${\displaystyle F(x)={\int }_{-\mathrm{\infty }}^{x}f(u)\mathrm{d}u,}$

${\displaystyle f(x)=\frac{\mathrm{d}}{\mathrm{d}x}F(x).}$

${\displaystyle f=\frac{\mathrm{d}{X}_{\ast }P}{\mathrm{d}\mu }.}$

${\displaystyle Pr[X\in A]={\int }_{X^{-1}A}\mathrm{d}P={\int }_{A}f\mathrm{d}\mu }$

${\displaystyle \mathrm{E}[X]={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}xf(x)dx.}$

${\displaystyle x\to F_X(x)=\mathrm{P}(X\le x)}$

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

${\displaystyle f(t)=\frac{1}{2}(\delta (t+1)+\delta (t-1)).}$

${\displaystyle f(t)=\sum _{i=1}^n{p}_i\delta (t-x_i),}$

${\displaystyle Pr\left(X_1,\dots ,X_N\in D\right)={\int }_D{f}_{X_1,\dots ,X_N}(x_1,\dots ,x_N)d{x}_1\cdots d{x}_N.}$

${\displaystyle f(x)=\frac{{\mathrm{\partial }}^{n}F}{\mathrm{\partial }{x}_1\cdots \mathrm{\partial }{x}_n}{|}_x}$

${\displaystyle f_{X_i}(x_i)=\int f(x_1,\dots ,x_n)d{x}_1\cdots d{x}_{i-1}d{x}_{i+1}\cdots d{x}_n.}$

${\displaystyle f_{X_1,\dots ,X_n}(x_1,\dots ,x_n)=f_{X_1}(x_1)\cdots f_{X_n}(x_n).}$

${\displaystyle f_{X_1,\dots ,X_n}(x_1,\dots ,x_n)=f_1(x_1)\cdots f_n(x_n),}$

${\displaystyle f_{X_i}(x_i)=\frac{f_i(x_i)}{\int f_i(x)dx}.}$

${\displaystyle Pr\left(X>0,Y>0\right)={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{f}_{X,Y}(x,y)dxdy.}$

${\displaystyle f_{U+V}(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_U(y)f_V(x-y)dy=\left(f_U\ast f_V\right)(x)}$

${\displaystyle f_{U_1+\dots +U_N}(x)=\left(f_{U_1}\ast \dots \ast f_{U_N}\right)(x)}$

${\displaystyle f_Y(y)=\left|\frac{1}{g^{\prime }(g^{-1}(y))}\right|\cdot f_X(g^{-1}(y)).}$

${\displaystyle \left|f_Y(y)dy\right|=\left|f_X(x)dx\right|,}$

${\displaystyle f_Y(y)=\left|\frac{dx}{dy}\right|f_X(x)=\left|\frac{1}{g^{\prime }(x)}\right|f_X(x)=\left|\frac{1}{g^{\prime }(g^{-1}(y))}\right|f_X(g^{-1}(y)).}$

${\displaystyle \sum _{k=1}^{n(y)}\left|\frac{1}{g^{\prime }(g_k^{-1}(y))}\right|\cdot f_X(g_k^{-1}(y))}$

${\displaystyle E(g(X))={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}y{f}_{g(X)}(y)dy,}$

${\displaystyle E(g(X))={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g(x)f_X(x)dx.}$

${\displaystyle g(\mathbf{y})=f(\mathbf{x})\left|det\left(\frac{d\mathbf{x}}{d\mathbf{y}}\right)\right|}$

${\displaystyle f_Y(y)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\dots {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{X_1}(x_1)f_{X_2}(x_2)\dots f_{X_n}(x_n)\delta (y-G(x_1,x_2,\dots x_n))d{x}_{1}d{x}_2\dots d{x}_n}$