概率论基础

1. 条件概率：\[P\left( {B|A} \right) = \frac{{P(A,B)}}{{P(A)}}\]
2. 乘法定理：\[\begin{array}{l}
   P\left( {A,B} \right) = P\left( {B|A} \right)P\left( A \right)\\
   P\left( {A,B,C} \right) = P\left( {C|A,B} \right)P\left( {B|A} \right)P\left( A \right)
   \end{array}\]
3. 全概率公式：\[P\left( A \right) = \sum\limits_j {P\left( {A|{B_j}} \right)P\left( {{B_j}} \right)} \]
4. 贝叶斯公式：\[P\left( {{B_i}|A} \right) = \frac{{P\left( {A|{B_i}} \right)P\left( {{B_i}} \right)}}{{\sum\limits_j {P\left( {A|{B_j}} \right)P\left( {{B_j}} \right)} }}\]
5. A，B独立：\[P\left( {A,B} \right) = P\left( A \right)P\left( B \right)\]
6. 概率分布函数与概率密度函数：\[\begin{array}{l}
   F\left( x \right) = P\left( {X \le x} \right) = \int_{ - \infty }^x {f\left( t \right)} \\
   f\left( x \right) = {F^`}\left( x \right)
   \end{array}\]
7. 期望：\[E\left( x \right) = \int_{ - \infty }^\infty  {xf\left( x \right)dx} \]
8. 方差：\[D\left( X \right) = E\left\{ {{{\left[ {X - E\left( X \right)} \right]}^2}} \right\} = E\left[ {{X^2}} \right] - {\left[ {E\left( X \right)} \right]^2}\]
9. 协方差：\[Cov\left( {X,Y} \right) = E\left\{ {\left[ {X - E\left( X \right)} \right]\left[ {Y - E\left( Y \right)} \right]} \right\} = E\left( {XY} \right) - E\left( X \right)E\left( Y \right)\]
10. 相关系数：\[{\rho _{XY}} = \frac{{Cov\left( {X,Y} \right)}}{{\sqrt {D\left( X \right)} \sqrt {D\left( Y \right)} }}\]