Random Variables

A random variable is a function from outcomes to values, not the realized value itself. Given a probability space $(\Omega ,\mathcal{F},P)$, a real-valued random variable is a measurable map $X:\Omega \to \mathbb{R}$. Its distribution is the induced probability law

$$P_X(B)=P(\{\omega \in \Omega :X(\omega )\in B\}).$$

This lets one talk about common distributions, expectation and variance, and relationships such as covariance and correlation without listing raw outcomes each time.

Intuition

The same outcome can feed many random variables. For two dice, one variable might be the sum, another the maximum, and another an indicator that the roll is a double. The distribution is determined by the mapping plus the underlying probability measure.

Worked computation

Let $X$ be the maximum of two fair dice. For $X=k$, at least one die must show $k$ and neither die can exceed $k$. The count is therefore $k^2-(k-1{)}^2=2k-1$ out of 36 ordered outcomes.

$k$	Count	$P(X=k)$
1	1	$1/36$
2	3	$3/36$
3	5	$5/36$
4	7	$7/36$
5	9	$9/36$
6	11	$11/36$

The expectation is

$$\mathbb{E}[X]=\frac{1\cdot 1+2\cdot 3+3\cdot 5+4\cdot 7+5\cdot 9+6\cdot 11}{36}=\frac{161}{36}\approx \mathrm{4.4722.}$$

Using the same probabilities, $\mathbb{E}[X^2]=791/36$, so $\mathrm{Var}(X)=791/36-(161/36{)}^2\approx 1.9715$. The random variable “maximum of two dice” is biased toward high values because many more outcomes map to 5 or 6 than to 1 or 2.

Caveats

Observed columns are samples from variables, not the variables themselves. A feature distribution can change under sampling bias or deployment drift, so claims about $X$ must name the population and measurement process.

References

• OpenStax Introductory Statistics 2e, Chapter 4 introduction
• Probability space: random variables

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