Independent Random Variables¶

Definition¶

Real-valued random variables $X$ and $Y$ are independent if

for every pair of intervals $A,B$; equivalently, for every pair of Borel sets $A,B$. For a finite family, mutual independence requires the analogous product formula for every selection of intervals (one for each variable) and every subfamily, rather than merely for pairs.

For discrete variables with a joint PMF, this is equivalent to $p_{X,Y}(x,y)=p_X(x)p_Y(y)$ for every $x,y$. For a pair with a joint density, it is equivalent to $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ for all $(x,y)$ except possibly on a set of zero area, for example along finitely many lines. In either case, conditional laws agree with the corresponding marginal laws wherever the conditioning denominator is positive (with density identities allowed to fail on a set of zero length or area).

Interpretation¶

Independent variables supply separate sources of randomness: learning one variable does not change the distribution of the other. PMF and PDF factorization are two ways to check the same property.

Every cell is the product of its row and column marginal, e.g. $\mathbb{P}(X=1,Y=1)=0.5\cdot0.6=0.30$$0.30$: $X$ and $Y$ are independent, $\mathbb{P}(X\in A,\,Y\in B)=\mathbb{P}(X\in A)\mathbb{P}(Y\in B)$ for all $A,B$; knowing $Y$ does not change the distribution of $X$.

The threshold events generated by the variables must be independent events for every selection of thresholds. This condition factors the joint law, rather than comparing only the marginal distributions or one pair of events.

Key Results¶

• For arbitrary real-valued $X,Y$, independence is equivalent to $F_{X,Y}(x,y)=F_X(x)F_Y(y)$ for every $x,y$.
• If $X,Y$ are independent and integrable, then $XY$ is integrable and $\mathbb{E}[XY]=\mathbb{E}X\,\mathbb{E}Y$$\mathbb{E}X\mathbb{E}Y$.

Examples¶

• Two independent fair dice have $p_{X,Y}(i,j)=1/36=(1/6)(1/6)$ for all $i,j\in\{1,\ldots,6\}$$\{1,\dots ,6\}$.
• The joint density $2$ on $0<x<y<1$ has marginals $f_X(x)=2(1-x)$ and $f_Y(y)=2y$ on $(0,1)$. At $(x,y)=(1/4,3/4)$ their product is $9/4$, rather than $2$, and this discrepancy persists on a whole neighbourhood, which has positive area; hence the density does not factor.

Prerequisites¶

• Random Variable
• Independent Events
• Distribution

Related Concepts¶

• Variance and Standard Deviation
• Conditional Probability Mass Function
• Joint CDF of Discrete Random Variables
• Conditional Probability Density Function
• Covariance
• Joint CDF of Continuous Random Variables
• Conditional Cumulative Distribution Function
• Correlation
• Joint Probability Mass Function
• Marginal Probability Mass Function
• Joint Probability Density Function
• Marginal Probability Density Function
• Binomial Distribution

Sources¶

• Probability Theory for EOR · Week 3
• Nicky van Foreest · Probability Distributions · Week 1
• Nicky van Foreest · Probability Distributions · Week 2

Aliases: independent r.v.s, independence of random variables, statistically independent random variables Keywords: independent variables, Independence of Discrete Random Variables, Independence of Continuous Random Variables Symbols: element of

Last updated: 22 October 2026 19:04