Probability Mass Function and Probability Density Function

The probability mass function (PMF) and probability density function (PDF) are two of the most important tools in statistics: they describe how probability is distributed across the possible values of a random variable. The distinction between them comes down to one thing — whether the variable is discrete or continuous. This guide explains what each function is, how they differ, and why the difference matters, building on the basics of probability. It's a core concept in quantitative qualifications like the FRM.

Discrete vs continuous variables

The starting point is the type of random variable you're dealing with:

• A discrete random variable can take only specific, separate values — the number on a die (1 to 6), the number of defaults in a loan portfolio, the count of trades in a day. You can list the possible values.
• A continuous random variable can take any value within a range — a share price, an interest rate, a return. Between any two values there are infinitely many others.

This difference is what gives rise to two distinct functions: the PMF for discrete variables and the PDF for continuous ones.

The probability mass function (PMF)

A probability mass function applies to a discrete random variable. It gives the probability that the variable takes exactly a particular value. For a fair die, the PMF says the probability of rolling a 3 is 1/6, and the same for every other face. Because it deals in specific outcomes, a PMF has two defining properties: each probability is between 0 and 1, and the probabilities of all possible values add up to 1. You can sensibly ask "what is the probability the value equals 3?" and read the answer straight off the PMF.

The probability density function (PDF)

A probability density function applies to a continuous random variable — and here there's a subtle but crucial twist. For a continuous variable, the probability of any single exact value is effectively zero, because there are infinitely many possible values. So a PDF doesn't give the probability of an exact value; instead it describes the relative density of probability across the range. Probability is found over an interval, as the area under the curve between two points. To get the probability that a return falls between, say, 2% and 5%, you measure the area under the PDF between those two values. The total area under the entire curve equals 1, mirroring the PMF's "sums to 1" rule.

The key difference, summarised

Both functions describe how probability is spread across a variable's possible values, and for both the total probability is 1. The difference is how you read them: a PMF gives the probability at a point (for discrete variables), while a PDF gives probability over an interval, as an area (for continuous variables). Mistaking the value of a PDF at a point for a probability is one of the most common errors students make — the height of a PDF is a density, not a probability.

Why it matters in finance

Financial quantities like returns, prices and interest rates are typically modelled as continuous random variables, so the PDF is central to finance. The famous bell-shaped normal distribution is described by a PDF, and risk measures such as value at risk are read as areas under a distribution's curve — the probability of a loss beyond a certain size. Discrete events, like the number of defaults in a portfolio, are modelled with a PMF. Understanding which function applies, and how to interpret it, is fundamental to risk modelling and quantitative analysis.

Frequently asked questions

What is the difference between a PMF and a PDF?

A PMF describes a discrete random variable and gives the probability of an exact value; a PDF describes a continuous random variable and gives probability over an interval, as the area under the curve. The PDF's value at a point is a density, not a probability.

Why is the probability of an exact value zero for a continuous variable?

Because a continuous variable has infinitely many possible values, the probability concentrated on any single exact point is effectively zero. Probability only becomes meaningful over a range.

Do both functions sum to 1?

Yes. For a PMF, the probabilities of all possible values add up to 1; for a PDF, the total area under the curve equals 1.

Why do PDFs matter in finance?

Financial variables like returns and prices are continuous, so they're modelled with PDFs. Risk measures such as value at risk are read as areas under a distribution's curve.

Build your quant skills with Learnsignal

PMFs and PDFs are the foundation of distributions and risk modelling. Learnsignal's tutor-led courses, including the FRM, develop the statistical understanding that topics like this build on — with clear teaching that turns abstract theory into something you can actually use.