What is a probability function?

A probability function is a mathematical relation that describes how probability is distributed over the possible outcomes of a random variable. It comes in two forms: the Probability Mass Function (PMF) for discrete variables, which assigns probability to each individual value, and the Probability Density Function (PDF) for continuous variables, where probabilities come from areas under the curve.

What is the difference between PMF and PDF?

The PMF (Probability Mass Function) is for discrete random variables and directly gives the probability that X equals a specific value: p(x) = P(X = x). The PDF (Probability Density Function) is for continuous variables and shows probability density rather than probability itself—actual probabilities come from integrating the PDF over an interval to get the area under the curve.

What properties must a valid probability function have?

A probability function must satisfy two essential properties: First, it cannot be negative—p(x) ≥ 0 or f(x) ≥ 0 for all x. Second, it must account for all probability—for discrete variables, the sum of all p(x) equals 1, and for continuous variables, the integral of f(x) over all real numbers equals 1.

How do you determine a probability function?

There are several ways: using symmetry for fair models (coin toss, die roll), using counting methods for combinatorial problems, adopting a standard distribution that fits the situation (binomial, normal, Poisson, etc.), or estimating from data by counting frequencies (empirical PMF) or building histograms and density estimates (empirical PDF).

Why do we need probability functions?

Probability functions are the foundation of probability theory. They connect random experiments to mathematical behavior, build probability distributions, enable calculation of expectations and variance, allow modeling of real-world phenomena, and answer questions about likelihood, ranges, and extreme events. Without them, we have only raw data, not predictive models.

Probability Function

Ways to Figure Out the Probability Function

The Two Forms of a Probability Function

Probability Mass Function (PMF)

Probability Density Function (PDF)

Notation and Symbols

Properties of a Probability Function

How We Use a Probability Function

What Questions It Answers

Why We Need It

Connection to Other Probability Concepts

The Engine of Probability

The probability function is the mathematical core of any random variable. It is the specific rule—whether a formula, a table, or a graph—that translates uncertain outcomes into precise numbers, telling us exactly how probability is distributed across the realm of possibilities.

Ways to Figure Out the Probability Function

A probability function doesn't appear automatically. There are only a few practical ways we decide what the probabilities (or densities) should be. Almost every situation fits into one of the following approaches.

1. Using Symmetry (Fair Models)
Sometimes the setup itself tells us all outcomes are equally likely.
Examples:
a fair coin, a fair die, a shuffled deck.
In these cases:
• every outcome gets the same chance
• the probability function spreads probability evenly across all possible values

2. Using Counting (Combinatorics)
When outcomes are equally likely but involve structure (drawing balls, cards, forming groups), we use counting.
We figure out the probability function by:
• counting how many outcomes match the condition
• counting how many total outcomes exist
• assigning probabilities using the ratio "favorable / total"

3. Using a Probabilistic Model (Standard Distributions)
Many real situations follow known patterns. We use a known distribution instead of deriving everything from scratch.
Examples:
• repeated independent trials → binomial distribution
• waiting times → exponential distribution
• measurement noise → normal distribution
• rare events in time → Poisson distribution

4. Using Data (Empirical Estimation)
When we have observations, we estimate the probability function directly from data.
For discrete data:
• count how often each value appears
• divide by the total number of observations
• this gives the empirical PMF

For continuous data:
• build a histogram or use smoothing to approximate the density
• the resulting curve acts as an estimated PDF

The Two Forms of a Probability Function

A probability function comes in two different forms, depending on whether the random variable takes separate values or ranges over a continuum. These two forms behave differently, but both serve the same purpose: they show how probability is spread out.

1. Probability Mass Function (PMF) — Discrete Case
The PMF is used when the random variable takes separate, countable values (like 0, 1, 2, 3… or the faces of a die).
• p(x) tells us the probability that X equals the value x.
• Each value gets its own probability.
• All the probability values together must add up to 1.
Examples: number of heads, number of arrivals, rolling a die, drawing a card from a finite deck.

2. Probability Density Function (PDF) — Continuous Case
The PDF is used when the random variable ranges over a continuous interval (like measurements, time, distance).
• f(x) is not a probability; it shows how tightly probability is packed around x.
• Actual probabilities come from the area under the curve of f(x).
• The total area under the entire density curve must be 1.
Examples: waiting times, heights, lengths, measurement errors, anything that can take any real value in an interval.

3. How They Relate
• Both PMF and PDF describe where probability is located.
• PMF handles individual points; PDF handles intervals.
• PMF assigns probabilities directly; PDF requires integration to get probabilities.
• They both reflect the same idea: how likely different outcomes are, given the type of random variable we're working with.

These two forms are simply two versions of the same concept — the probability function — adapted to the nature of the random variable.

Probability Mass Function (PMF)

The Probability Mass Function (PMF) serves as the fundamental building block for understanding discrete random variables. Unlike continuous distributions where probability spreads across intervals, discrete random variables can only take on specific, countable values—like the number of heads in coin flips or the sum when rolling dice.

The PMF, denoted

p(x)

P(X = x)

, tells us exactly how much probability "mass" sits at each possible outcome. Think of it like distributing weight across specific points rather than spreading it continuously. For a fair six-sided die, the PMF assigns exactly

\frac{1}{6}

probability mass to each outcome from 1 to 6.

Two critical properties define valid PMFs: First, probabilities must be non-negative:

p(x) \geq 0

for all

x

. Second, the total probability must equal one:

\sum_{\text{all } x} p(x) = 1

. This captures the certainty that *something* must occur when we observe the random variable.

Visualizing a PMF typically involves vertical bars or "impulses" at each possible value, with heights representing probabilities. This distinct, separated structure contrasts sharply with the smooth curves of continuous probability density functions. The PMF gives us complete probabilistic information—knowing it lets us calculate probabilities for any event involving our discrete random variable, whether we're finding

P(X > 3)

or computing expected values and variances.

Interactive visualization of six fundamental discrete distributions

Discrete Uniform

Equal probability for finite outcomes

Minimum Value (a): 1

Maximum Value (b): 6

Explanation

A discrete uniform distribution assigns equal probability to each value in a finite range. The probability mass function is $P(X = k) = \frac{1}{b - a + 1}$ for $a \leq k \leq b$ . The expected value is $E[X] = \frac{a + b}{2}$ , and the variance is $\text{Var}(X) = \frac{n^2 - 1}{12}$ , where $n = b - a + 1$ . Common examples include rolling a fair die, selecting a random card from a deck, or generating a random number from a finite range.

Probability Density Function (PDF)

The Probability Density Function (PDF) is how we work with continuous random variables—things that can be any real number in a range, not just discrete counts. Think measurements: weight, voltage, reaction time.

Here's the core problem: if a variable can be literally any value between 0 and 10, there are infinitely many possibilities. So asking "what's the probability it equals exactly 7.3528492...?" gives you zero. With infinite options, any single point has zero probability.

The PDF

f(x)

solves this by measuring probability concentration rather than probability itself. Where

f(x)

is high, values cluster. Where it's low, they're sparse. But

f(x) = 0.8

doesn't mean "80% probability"—it means that region has high density compared to other regions.

To get actual probabilities, you need intervals. Integration gives you the area under the curve:

P(a \leq X \leq b) = \int_{a}^{b} f(x)\,dx

This area represents the actual probability of landing between

a

and

b

.

For any legitimate PDF: first,

f(x) \geq 0

always (negative density makes no sense), and second, the total area under the entire curve must equal 1 since the variable has to take *some* value:

\int_{-\infty}^{\infty} f(x)\,dx = 1

The takeaway: PDFs show you where probability concentrates through their shape. Tall peaks mean common values. Only areas—not heights—give you probabilities.

Interactive visualization of fundamental continuous distributions

Continuous Uniform

Constant probability over an interval

Lower Bound (a): 0

Upper Bound (b): 10

Explanation

The continuous uniform distribution has constant probability density over the interval $[a, b]$ . The probability density function is $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$ , and $0$ otherwise. The expected value is $E[X] = \frac{a+b}{2}$ and the variance is $\text{Var}(X) = \frac{(b-a)^2}{12}$ . This distribution models situations where all values in an interval are equally likely, such as the position of a randomly thrown dart on a board, random arrival times within a time window, or measurement errors uniformly distributed within tolerances.

Notation and Symbols

When we work with probability functions, a few symbols show up again and again.
Here is what they really mean in everyday language:

(X)

— this is the random variable. Think of it as the "thing that can happen" when you run a random experiment: the number on a die, waiting time, measurement, etc.

(x)

— a particular outcome or value that (X) might take. It's just one possible result you might see.

(p(x))

— what the probability function looks like in the discrete case. It tells you directly: *"What is the chance that the outcome is exactly (x)?"

(f(x))

— what the probability function looks like in the continuous case. It's not the probability of (X = x) (that's zero for continuous variables). Instead, it shows how "dense" the probability is around that point — where the curve rises or falls.

(P(,\cdot,))

— our general way to talk about the probability of something happening.
Whatever we put inside the parentheses describes the event:

(P(X=3))

(P(X>10))

(P(a \le X \le b))

, etc.

(\int f(x),dx)

— the tool we use to get actual probabilities in the continuous case.
It measures the area under the curve over an interval, which *is* the probability for continuous variables.

This is the "language" we use to describe how probabilities behave, whether the outcomes are separate points or part of a continuous range.

See All Probability Symbols and Notations →

Properties of a Probability Function

A probability function (whether it is a PMF or a PDF) has to follow a few basic requirements to make sense. These describe what the function itself must look like and how it must behave.

1. It Cannot Be Negative
• For discrete variables: p(x) ≥ 0 for all x
• For continuous variables: f(x) ≥ 0 for all x
A probability or density can never go below zero.

2. It Must Account for All Probability
• For discrete variables: the sum of p(x) over all possible x equals 1
• For continuous variables: the total area under f(x) over the entire real line equals 1
Together, all possible outcomes must absorb all the probability.

There are other characteristics that can describe how a probability function behaves, but these two are the essential ones. Everything else you'll ever see—how intervals get probabilities, how events combine, how distributions are shaped—comes from the basic axioms of probability and the conclusions that follow from them. These two conditions are the foundation, and the rest grows naturally from there.

How We Use a Probability Function

Once we know the probability function of a random variable, we can start using it to answer meaningful questions. The probability function is the tool that lets us turn the idea of uncertainty into concrete numbers.

1. Finding the Probability of an Event
• For discrete variables, we add up p(x) for the values that belong to the event.
• For continuous variables, we take the area under f(x) over the interval of interest.
This tells us how likely a certain outcome or range of outcomes is.

2. Understanding the Shape of a Distribution
• The probability function shows where probability is concentrated.
• It helps us see if values cluster, spread out, or have long tails.
• It tells us whether outcomes tend to be centered around something or scattered widely.

3. Computing the Expected Value and Variance
• The expected value uses p(x) or f(x) to find the long-run average outcome.
• The variance shows how much the values tend to vary around that average.
These are key numbers for summarizing the behavior of a random variable.

4. Making Predictions and Decisions
• Probability functions help model real processes, such as waiting times, counts, or measurements.
• They let us estimate risks, compare scenarios, and evaluate chances of rare or extreme events.
• They are central to simulations, forecasting, and probability-based decision making.

A probability function is more than a definition — it is the engine behind almost every calculation and interpretation in probability. Once we have it, we can analyze the random variable in a meaningful, quantitative way.

What Questions It Answers

A probability function is a diagnostic tool. Once defined, it allows us to interrogate the random variable and get precise answers about its behavior:

* Likelihood: Which specific outcomes are most frequent?
* Range: What is the exact probability that the result falls between

a

and

b

?
* Shape: Is the randomness symmetric, or does it lean heavily in one direction (skewed)?
* Extremes: How likely are rare, extreme events (the "tails" of the distribution)?

Why We Need It

The probability function is the bridge between the physical world of experiments and the theoretical world of mathematics.

* It Builds the Distribution: You cannot mathematically define a probability distribution without the function that governs it.
* It Enables Calculation: Every metric we care about—Expectation (averages), Variance (risk), and Quantiles—is calculated directly from this function.
* It Allows Modeling: To analyze real-world phenomena (like stock prices, defect rates, or waiting times), we must fit a probability function to the reality. Without it, we have only raw data, not a predictive model.

Connection to Other Probability Concepts

The probability function does not stand alone. It connects directly to other major ideas in probability, forming the backbone of the entire subject.

Random Variables
• The probability function describes how a random variable behaves.
• Once we know the PF, we fully understand the variable's distribution.

Distributions
• Every probability distribution is built from its probability function.
• The PF tells us how probability is spread out; the distribution summarizes that spread.

The Cumulative Distribution Function (CDF)
• The CDF adds up probability from negative infinity up to a value.
• It is built directly from the PF:
– by summing the PMF in the discrete case,
– by integrating the PDF in the continuous case.

4. Probability Models
• A probability model combines:
-the sample space,
-the events,
-and the probability function.
• Without the PF, a probability model cannot assign chances to outcomes.

5. Real-World Problems
• Almost every calculation—predicting outcomes, estimating risks, modeling processes—begins with knowing the PF.
• It is the starting point for simulations, decision-making, and statistical inference.

The probability function is not just a definition. It is the link that connects random variables, distributions, cumulative functions, and probability models into one coherent framework.