Numerical Characteristics of Random Variables

This chapter systematically studies the distribution of random variables, mathematical expectation, variance, covariance, correlation coefficient, and other numerical characteristics.

Random Variables and Distributions

• Random variable: a variable whose value is uncertain
• Probability law, distribution function

Mathematical Expectation

• Discrete: $E(X) = \sum x_i P(X=x_i)$
• Continuous: $E(X) = \int_{-\infty}^{+\infty} x f(x) dx$
• Properties: linearity, expectation of a constant, etc.

Variance and Standard Deviation

• $D(X) = E[(X-E(X))^2]$
• $\sigma = \sqrt{D(X)}$
• Property: $D(aX+b) = a^2 D(X)$

Covariance and Correlation Coefficient

• Covariance: $Cov(X,Y) = E[(X-E(X))(Y-E(Y))]$
• Correlation coefficient: $\rho_{XY} = \frac{Cov(X,Y)}{\sigma_X \sigma_Y}$

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Exercises

1. Given the probability law of $X$: $P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3$, find $E(X)$.
2. Given $E(X)=2, D(X)=3$, find $E(3X-1)$ and $D(3X-1)$.
3. Given $X, Y$ are independent, $E(X)=1, E(Y)=2$, find $E(X+Y)$.
4. Given $E(X)=1, E(Y)=2, Cov(X,Y)=3, \sigma_X=2, \sigma_Y=1$, find $\rho_{XY}$.
5. Given the distribution function $F(x)=\begin{cases} 0, x<0 \\ x, 0\leq x<1 \\ 1, x\geq1 \end{cases}$, find $E(X)$.

Reference Answers

1. $E(X)$

$1\times0.2+2\times0.5+3\times0.3=0.2+1+0.9=2.1$

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2. $E(3X-1)$ and $D(3X-1)$

$E(3X-1)=3E(X)-1=3\times2-1=5$

$D(3X-1)=9D(X)=9\times3=27$

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3. $E(X+Y)$

$E(X)+E(Y)=1+2=3$

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4. $\rho_{XY}$

$\frac{3}{2\times1}=1.5$

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5. $E(X)$

$E(X)=\int_0^1 x dx=\frac{1}{2}$

上一章节 Random Events and Probability Understand the basic concepts of random events and probability, and master probability calculation formulas and event independence. 下一章节 Parameter Estimation Learn the basic ideas and methods of point estimation and interval estimation, such as the method of moments and the method of maximum likelihood.