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. $E(X)$

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