Raising the bar

Chapter 6. Continuous Random Variables and Probability Distributions

Unit 6.8. Joint distribution and independence

All definitions in this section have precedents.

Joint and marginal distribution functions. Consider vectors $latex (X,Y)$ and $latex (x,y)$ ($latex (X,Y)$ is a pair of random variables and $latex (x,y)$ is a pair of real numbers). The function $latex {F_{(X,Y)}}(x,y) = P(X \le x,Y \le y)$ is called a joint distribution function of the vector $latex (X,Y)$. The marginal distribution functions are defined to be $latex {F_{(X,Y)}}(x,\infty),\,\,{F_{(X,Y)}}(\infty ,y)$.

Exercise 6.4. What is the relationship between marginal distribution functions and "own" distribution functions $latex F_X(x),\ F_Y(y)$?

Joint density and independence. We say that a function $latex {p_{(X,Y)}}({t_1},{t_2})$ is a joint density of $latex (X,Y)$ if

$latex {F_{(X,Y)}}(x,y) = \int\limits_{ - \infty }^x \int\limits_{ - \infty }^y {p_{(X,Y)}}({t_1},{t_2})d{t_1}d{t_2} \textrm{ for all real } x,y.$

Suppose $latex X$ and $latex Y$ have densities $latex {p_X}$ and $latex {p_Y}$, respectively. $latex X$ and $latex Y$ are called independent if the joint density decomposes into a product of own densities: $latex {p_{(X,Y)}}({t_1},{t_2}) = {p_X}({t_1}){p_Y}({t_2})$.

Exercise 6.5. For independent continuous random variables we have multiplicativity of means: $latex EXY = (EX)(EY)$.

Proof. Everything follows from definitions:

$latex E(XY) = \int\limits_{\mathbb{R}}^{} \int\limits_{\mathbb{R}}^{} {{t_1}{t_2}{p_{(X,Y)}}({t_1},{t_2})d{t_1}d{t_2}}$ (by independence)

$latex = \int\limits_{\mathbb{R}}^{} \int\limits_{\mathbb{R}}^{} {{t_1}{t_2}{p_X}({t_1}){p_Y}({t_2})d{t_1}d{t_2}}$

$latex = \int\limits_{\mathbb{R}}^{} {{t_1}{p_X}({t_1})d{t_1}} \int\limits_{\mathbb{R}}^{} {{t_2}{p_Y}({t_2})d{t_2}} = (EX)(EY).$

Constructing independent standard normal variables. We wish to have independent standard normal variables $latex {z_1},{z_2}$ (both must have the same density $latex {p_z}(t)$). To satisfy the definition of independence put $latex {p_{({z_1},{z_2})}}({t_1},{t_2}) = {p_{{z_1}}}({t_1}){p_{{z_2}}}({t_2})$. In a similar way for any natural $latex n$, we can construct $latex n$ independent standard normal variables.

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