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mathematics_basic_for_ML
  • README
  • README
    • Summary
    • Geometry
      • EulerAngle
      • Gimbal lock
      • Quaternion
      • RiemannianManifolds
      • RotationMatrix
      • SphericalHarmonics
    • Information
      • Divergence
      • 信息熵 entropy
    • LinearAlgebra
      • 2D仿射变换(2D Affine Transformation)
      • 2DTransformation
      • 3D变换(3D Transformation)
      • ComplexTransformation
      • Conjugate
      • Hessian
      • IllConditioning
      • 逆变换(Inverse transform)
      • SVD
      • det
      • eigendecomposition
      • 矩阵
      • norm
      • orthogonal
      • special_matrix
      • trace
      • vector
    • Mathematics
      • Complex
      • ExponentialDecay
      • average
      • calculus
      • convex
      • derivative
      • 距离
      • function
      • space
      • Formula
        • euler
        • jensen
        • taylor
        • trigonometric
    • Numbers
      • 几何级数
      • SpecialNumbers
    • NumericalComputation
      • ConstrainedOptimization
      • GradientDescent
      • Newton
      • Nominal
      • ODE_SDE
      • Preprocessing
    • Probability
      • bayes
      • distribution
      • expectation_variance
      • 贝叶斯公式
      • functions
      • likelihood
      • mixture_distribution
      • 一些术语
      • probability_distribution
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  • 条件概率的链式法则
  • 独立

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  2. Probability

probability_distribution

Previous一些术语

Last updated 2 years ago

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离散型变量
连续性变量

概率分布

概率质量函数

概率密度函数

性质1

定义域必须是x所有可能的状态的集合

同

性质2

Note:不要求

性质3(归一化)

部分概率

P(x = a)

联合分布

P(X=x, Y=y)

边缘概率

条件概率

P()为连续型变量落入某一区间的概率

条件概率的链式法则

P(A,B,C)=P(A∣B,C)P(B∣C)P(C)P(A, B, C) = P(A|B,C)P(B|C)P(C)P(A,B,C)=P(A∣B,C)P(B∣C)P(C)

独立

相互独立:P(A, B) = P(A)P(B) 条件独立:P(A, B | C) = P(A | C)P(B | C)

x∼P(x)x \sim P(x)x∼P(x)
x∼p(x)x \sim p(x)x∼p(x)
∀x,0≤P(x)≤1\forall x, 0 \leq P(x) \leq 1∀x,0≤P(x)≤1
∀x,0≤p(x)\forall x, 0 \leq p(x)∀x,0≤p(x)
p(x)≤1p(x) \leq 1p(x)≤1
∑a∈xP(a)=1\sum_{a \in x}P(a) = 1∑a∈x​P(a)=1
∫p(x)dx=1\int p(x)dx = 1∫p(x)dx=1
∫abp(x)dx\int_a^b p(x)dx∫ab​p(x)dx
∀a∈x,P(x=a)=∑b∈yP(x=a,y=b)\forall a \in x, P(x=a)=\sum_{b \in y}P(x=a, y=b)∀a∈x,P(x=a)=∑b∈y​P(x=a,y=b)
p(x)=∫p(x,y)dyp(x)=\int p(x,y)dyp(x)=∫p(x,y)dy
P(y=b∣x=a)=P(y=b,x=a)P(x=a)P(y=b\mid x=a) = \frac {P(y=b, x=a)}{P(x=a)}P(y=b∣x=a)=P(x=a)P(y=b,x=a)​
P(A∣B)=P(A,B)P(B)P(A\mid B) = \frac{P(A,B)}{P(B)}P(A∣B)=P(B)P(A,B)​