Numerical Characteristics of Random Variables
Expectation, Variance, Covariance & Correlation
1. Expectation & Expectation Operator
Expectation $E[X]$ is the weighted average of random variable values, representing the central position of the distribution.
$$E[X] = \int_{-\infty}^{\infty} x f(x) dx \quad \text{或} \quad \sum x_i p_i$$
Operator Properties:
- Linearity: $E[aX + bY] = aE[X] + bE[Y]$
- Independent Product: If $X,Y$ are independent, then $E[XY] = E[X]E[Y]$
2. Variance & Standard Deviation
Variance $D[X]$ describes the degree of deviation from the expectation, i.e., dispersion.
$$D[X] = E[(X - E[X])^2] = E[X^2] - (E[X])^2$$
$$\sigma(X) = \sqrt{D[X]}$$
3. Covariance & Correlation Coefficient
Covariance: Measures the direction of association between $X, Y$.
$$Cov(X, Y) = E[(X-E[X])(Y-E[Y])]$$
Correlation Coefficient: Linear correlation after eliminating dimensionality.
$$\rho_{XY} = \frac{Cov(X, Y)}{\sigma_X \sigma_Y} \in [-1, 1]$$
Scatter Feature Real-time Mapping
均值向量 $E[X], E[Y]$
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标准差 $\sigma_X, \sigma_Y$
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协方差 $Cov(X,Y)$
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相关系数 $\rho_{XY}$
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