Random variables and distribution functions

What is a random variable?

A random variable is a rule that gives every outcome of an uncertain event a number. In finance it is often called a stochastic variable . The number lets you do arithmetic on an uncertain future: add it up, average it, ask how likely a range of values is.

There are two kinds, and the kind decides which tool describes the probabilities.

• A discrete random variable has a countable set of distinct outcomes: the number of defaults in a loan book, or the number of quarters in which a fund beats its benchmark.
• A continuous random variable can take any value in a range: tomorrow’s index return, or the share of a loan recovered after a default.

Real data blur the line. A daily return rounded to a basis point can only take certain values, yet with many observations it is modeled as continuous. Continuous models are easier to work with, fit data well and give clean formulas.

Random variablea number for each outcome of an uncertain event

• Discrete Memorize countable outcomes, such as a count of defaults
  • Probability mass function (PMF) P(X = x) for each outcome
  • Cumulative distribution function (CDF) a staircase that jumps at each outcome
• Continuous Memorize any value in a range, such as a return
  • Probability density function (PDF) probability is the area under it
  • Cumulative distribution function (CDF) a smooth curve

How does the probability mass function work?

For a discrete variable, the probability mass function (PMF) lists P(X = x) for every outcome. The probabilities add to 1, and the probability of a range is the sum over the outcomes in it.

$$\sum_x P(X = x) = 1, \qquad \int_{-\infty}^{\infty} f(x)\,dx = 1$$

$$P(a \le X \le b) = \sum_{x=a}^{b} P(X = x)$$

Take X to be the number of the next four quarters in which a fund beats its benchmark. The second row below is the PMF. The third row is a running total, which the next section explains.

x	0	1	2	3	4
P(X = x)	0.09	0.16	0.44	0.24	0.07
F(x) = P(X ≤ x)	0.09	0.25	0.69	0.93	1.00

How does the cumulative distribution function work?

The cumulative distribution function (CDF) gives F(x) = P(X ≤ x). It never falls, starts at 0 far to the left and reaches 1 far to the right. It works for discrete and continuous variables alike.

$$F(x) = P(X \le x)$$

In words, F(2) = 0.69 in the fund table means a 69% chance that the fund beats its benchmark in two quarters or fewer.

For a discrete variable the CDF is a staircase: it jumps at each outcome by that outcome’s probability and is flat in between. That has a consequence for ranges. When both end points count, subtract F(a − 1), the step just below a, and not F(a).

Exam tip With a discrete table, say the range in words first: “from one up to and including three”. If the lower end is included, go one step below it.

How does the probability density function work?

For a continuous variable, probability is the area under the probability density function (PDF) between two points. That area equals F(b) − F(a). The PDF is the slope of the CDF, and its height is a density, not a probability.

$$P(a \le X \le b) = \int_a^b f(x)\,dx = F(b) - F(a)$$

$$f(x) = \dfrac{dF(x)}{dx}$$

Here is a toy model of a recovery rate X between 0 and 1. Its CDF is F(x) = x², so its density is f(x) = 2x. You never need to do the calculus: you subtract two CDF values.

Start on the Discrete tab and drag the two range handles to see the bars that count and the jump they make in the cumulative staircase. Switch on the trap to see the wrong figure beside the right one. The Continuous tab shades the area, and the third tab shows the law of large numbers, which comes next.

Go deeper Why a density can be above 1

The total area under a PDF is 1, but the height can be larger than 1 where the variable is bunched up. In the recovery-rate model the density is f(0.9) = 1.8, yet no probability is above 1: probabilities are areas, and a narrow strip at height 1.8 has a small area. Never read a density height as “the chance of that value”.

Go deeper Why finance leans on continuous models

Prices move in ticks, so every price is technically discrete. Analysts still model returns as continuous because the formulas are cleaner, a smooth curve fits many observations well, and tools such as the normal distribution come with ready-made results. The next lessons use both kinds: counts (defaults, up moves) are discrete, returns and rates are continuous.

What does the law of large numbers say?

The law of large numbers says that as the number of independent trials grows, the sample mean gets closer and closer to the true (unconditional) mean, provided the sample is random and representative. It is why a long history or a long simulation estimates an expected value well.

Do not confuse it with the central limit theorem. The law of large numbers says where the sample mean ends up: at the true mean. The central limit theorem, covered in Learning Module 7, says what shape the sample mean has: approximately normal. Exam options often pair them as distractors.