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 ProbabilityUnderstand the basic concepts of random events and probability, and master probability calculation formulas and event independence. 下一章节 Law of Large Numbers and Central Limit TheoremUnderstand Chebyshev's inequality, the law of large numbers, and the central limit theorem and their applications.