Raising the bar

Chapter 6. Continuous Random Variables and Probability Distributions

Unit 6.4. Mean, covariance, variance and correlation

It might be relieving to know that it's only the definition of the mean that is different from the discrete case. All properties of means, covariance and variance derived from that definition are absolutely the same as in the discrete case. Here is the list of properties that will be constantly used:

1. Linearity of means: $latex E(aX + bY) = aEX + bEY$.
2. Expected value of a constant: $latex Ec = c$.
3. Multiplicativity of expected values: $latex EXY = (EX)(EY)$ for independent $latex X,\,Y$.
4. Linearity of covariance: for example, for the first argument we have $latex {\mathop{\rm cov}} (aX + bY,Z) = a{\mathop{\rm cov}} (X,Z) + b{\mathop{\rm cov}} (Y,Z).$
5. Shortcut: $latex {\mathop{\rm cov}} (X,Y) = EXY - (EX)(EY)$.
6. Independent variables are uncorrelated: $latex {\mathop{\rm cov}} (X,Y) = 0$.
7. Correlation with a constant: $latex {\mathop{\rm cov}} (X,c) = 0.$
8. Symmetry of covariance: $latex {\mathop{\rm cov}} (X,Y) = {\mathop{\rm cov}} (Y,X)$.
9. Relation to covariance: $latex V(X) = {\mathop{\rm cov}} (X,X)$.
10. Shortcut: $latex V(X) = E{X^2} - {(EX)^2}$.
11. Variance of a linear combination: $latex V(aX + bY) = {a^2}V(X) + 2ab{\mathop{\rm cov}} (X,Y) + {b^2}V(Y).$
12. Variance of a constant: $latex V(c) = 0$.
13. Nonnegativity: $latex V(X) \ge 0$.
14. Variables with vanishing variance: $latex V(X) = 0$ if and only if $latex X = c$.
15. Additivity of variance: $latex V(X + Y) = V(X) + V(Y)$ for independent $latex X,\,Y$.
16. Standard deviation is homogeneous: $latex \sigma (aX) = |a|\sigma (X)$.
17. Range of the correlation coefficient: $latex -1 \le \rho (X,Y) \le 1$ for any $latex X,Y$.
18. Standardization: for any variable $latex X$ with nonzero standard deviation, the variable $latex Y = \frac{{X - EX}}{{\sigma (X)}}$ has mean zero and unit variance.

In fact, some proofs require additional definitions: joint density, independence etc. We shall directly apply only the definition of a function of a random variable, and this is better explained following the cow-and-goat principle. Table 6.1 defines a function of a continuous random variable by way of similarity.

Table 6.1. Definition of a function of a random variable.

Example 6.1. $latex {X^2}$ takes values $latex {t^2}$. $latex \sin X$ takes values $latex \sin t$. Therefore

$latex E{X^2} = \int\limits_{ - \infty }^{ + \infty } {{t^2}{p_X}(t)dt},\,\,E\sin X = \int\limits_{ - \infty }^{ + \infty } {(\sin t){p_X}(t)dt.}$

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