Random Variables

Motivation: why this matters in quant finance

A probability space contains abstract outcomes. A random variable turns those outcomes into numbers: a terminal stock price, a log return, a default indicator, a portfolio loss, or an option payoff. Without random variables, a probability model can say which state occurred but cannot produce the numerical quantities quants price, hedge, forecast, and stress.

Bertsekas introduces random variables by asking how probabilities of numerical values are induced from probabilities of underlying outcomes. That direction matters. The model does not start with a histogram floating in space; it starts with outcomes $\omega$ , then a function $X(\omega)$ , then probabilities such as $\mathbb{P}(X=x)$ or $\mathbb{P}(X\le x)$ .

In finance, this distinction keeps three ideas separate: the state of the world

\omega

, the observable quantity

S_T(\omega)

, and the distribution of

S_T

. Two models may have the same terminal distribution but different path structure, which matters immediately for barriers, Asians, hedging, and filtrations.

The informal idea

A random variable is a measurement rule. It looks at the realised outcome and reports a number. If $\omega$ is the complete future market path, then

$S_T(\omega)$ reports the terminal stock price,
$(S_T(\omega)-K)^+$ reports the call payoff,
$\mathbf{1}_{\{S_T>K\}}(\omega)$ reports whether a digital call pays,
$\log(S_T(\omega)/S_0)$ reports the log return.

The distribution of $X$ forgets the identity of the original outcome and remembers only how much probability lands on each numerical value or interval. That loss of detail is useful, but it is also dangerous: a terminal distribution alone does not tell you what happened along the way.

Formal definitions

Random variable

Let

(\Omega,\mathcal{F},\mathbb{P})

be a probability space. A real-valued random variable is a measurable function

X:\Omega\to\mathbb{R}.

Measurability means

\{\omega\in\Omega:X(\omega)\le x\}\in\mathcal{F}\qquad\text{for every }x\in\mathbb{R}.

Equivalently, $X^{-1}(B)\in\mathcal{F}$ for every Borel set $B\subseteq\mathbb{R}$ . The shorter condition with half-lines is enough because half-lines generate the Borel sigma-algebra.

Distribution and CDF

The distribution or law of

X

is the probability measure on

\mathbb{R}

induced by

X

\mu_X(B)=\mathbb{P}(X\in B).

The cumulative distribution function is

F_X(x)=\mathbb{P}(X\le x).

Bertsekas uses the CDF as the common language for discrete and continuous random variables: every random variable has a CDF, even when it has neither a convenient PMF nor a smooth PDF.

Discrete and continuous cases

If $X$ takes countably many values $x_i$ , its PMF is

p_X(x_i)=\mathbb{P}(X=x_i),\qquad \sum_i p_X(x_i)=1.

If $X$ has a density $f_X$ , then

F_X(x)=\int_{-\infty}^{x} f_X(u)\,du.

Key properties

Random variables push probability forward

The event $\{X\in B\}$ is a subset of $\Omega$ , but the distribution $\mu_X(B)$ is a probability on numerical sets $B\subseteq\mathbb{R}$ . This is why the same abstract probability space can support many financial quantities at once.

The CDF characterises the distribution

Knowing $F_X(x)$ for every $x$ determines the law of $X$ . For integer-valued $X$ ,

p_X(k)=F_X(k)-F_X(k-1).

For continuous $X$ with differentiable CDF,

f_X(x)=F_X'(x).

Functions of random variables remain random variables

If $Y=g(X)$ for a measurable function $g$ , then $Y$ is a random variable. Payoffs are built this way: $H=g(S_T)$ . The law of $Y$ is derived from the law of $X$ plus the transformation $g$ .

Several random variables require joint structure

Marginal distributions of $X$ and $Y$ do not determine the joint distribution. A two-asset option needs the joint law of $(S_T^{(1)},S_T^{(2)})$ , not only the two separate terminal distributions. Dependence enters through the joint law.

Equality has multiple meanings

Random variables can be equal pointwise, equal almost surely, or equal in distribution. In probability and finance, almost-sure equality is usually the operational notion: changing a payoff on a null event does not change its price.

Worked examples

Example 1: terminal stock price and payoff

In a one-period model with $\Omega=\{u,d\}$ ,

S_T(u)=110,\qquad S_T(d)=90.

For a call with strike $100$ ,

H(\omega)=(S_T(\omega)-100)^+,

so $H(u)=10$ and $H(d)=0$ . The payoff is a random variable derived from the price random variable.

Example 2: a CDF from a discrete PMF

Let $X$ be the number of heads in two fair coin tosses. Then

p_X(0)=\frac14,\qquad p_X(1)=\frac12,\qquad p_X(2)=\frac14.

Its CDF is a step function:

F_X(x)= \begin{cases} 0, & x<0,\\ 1/4, & 0\le x<1,\\ 3/4, & 1\le x<2,\\ 1, & x\ge2. \end{cases}

The jumps of the CDF are the point probabilities.

Example 3: why the joint law matters

Suppose two assets each have return $+10\%$ or $-10\%$ with probability $1/2$ . If they always move together, an equally weighted portfolio has return $+10\%$ or $-10\%$ . If one always rises when the other falls, the same portfolio has return $0$ with certainty. The marginal distributions are identical; the joint distribution changes the risk completely.

Common confusions and pitfalls

"A random variable is random because its value changes by itself." The function

X

is fixed. Randomness comes from the realised outcome

\omega

"The distribution is the same thing as the random variable." The distribution forgets the underlying state space. That is enough for vanilla terminal payoffs, but not for path-dependent products or conditioning on information.

"Every function of $\omega$ is allowed." It must be measurable. In ordinary quant examples this is rarely a practical obstacle, but it is the condition that makes probabilities like

\mathbb{P}(X\le x)

meaningful.

"Equal in distribution means interchangeable." Not for joint questions. Two variables can have the same law but different dependence with other variables.

"The PDF is more fundamental than the CDF." The CDF exists for every real-valued random variable. PDFs exist only for continuous laws with suitable smoothness.

Where this goes next

Expectation and Variance: Averages and dispersion are defined for random variables, not for abstract outcomes directly.
Independence and Conditioning: Joint laws, conditional laws, and information updates require random variables.
Sigma-Algebras: Measurability is the link between random variables and event collections.
Normal Distribution: The central continuous law for log returns and Brownian increments.
Log-Normal Distribution: The distribution of prices when log returns are normal.

References

Bertsekas, D. P., & Tsitsiklis, J. N. (2008). Introduction to Probability (2nd ed.). Athena Scientific. Ch. 2 §2.2 (Probability Mass Functions), §2.3 (Functions of Random Variables), Ch. 3 §3.2 (Cumulative Distribution Functions).