Probability Density/Mass Functions (PDF/PMF)

When analyzing data, we often need to quantify the likelihood of different outcomes. For example, what is the chance a customer makes 3 purchases in a month, or that a sensor reading falls between 20 and 25 degrees Celsius? The tools for answering these questions depend on whether the variable is discrete (countable outcomes) or continuous (outcomes within a range). Probability Mass Functions (PMFs) and Probability Density Functions (PDFs) provide the mathematical framework for this quantification.

Probability Mass Function (PMF) for Discrete Variables

A Probability Mass Function (PMF) describes the probability that a discrete random variable takes on a specific value. For any given outcome $x$ , the PMF, denoted as $P(X=x)$ or $f_X(x)$ , directly gives the probability of observing that outcome. This means $f_X(x)$ must be between 0 and 1, inclusive, for all possible values of $x$ . The sum of all probabilities for all possible outcomes must equal 1, representing the certainty that one of the possible outcomes will occur.

📐 PMF Properties

For a discrete random variable $X$ with possible values $x_1, x_2, \dots, x_n$ :

$0 \leq f_X(x_i) \leq 1$ for all $i$
$\sum_{i=1}^{n} f_X(x_i) = 1$

pythonCalculating PMF for Coin Flips

PMF for Number of Heads in 3 Coin Flips

This bar chart visualizes the Probability Mass Function for the number of heads observed in three independent coin flips. The x-axis represents the discrete number of heads (0, 1, 2, or 3), and the y-axis shows the probability of observing that specific number of heads. Notice the symmetric distribution, with 1 and 2 heads being the most probable outcomes.

Loading chart...

Key Insight: The height of each bar directly represents the probability of that specific discrete outcome.

Check Your Understanding

If a PMF for a variable

X

has

P(X=1)=0.4

and

P(X=2)=0.3

, what can we say about

P(X=3)

if these are the only possible outcomes?

Probability Density Function (PDF) for Continuous Variables

For continuous random variables, we use a Probability Density Function (PDF), denoted as $f_X(x)$ . Unlike PMFs, a PDF does not give the probability of a specific value $x$ . For any continuous variable, the probability of observing exactly one specific value is infinitesimally small, effectively zero. Instead, the PDF describes the relative likelihood for the variable to take on a given value within a continuous range. Higher values of $f_X(x)$ indicate a greater density of probability around $x$ .

⚠️ PDF Values Are Not Probabilities

It is crucial to understand that $f_X(x)$ itself is not a probability. A PDF value can even be greater than 1, which is impossible for a probability. Probabilities for continuous variables are found by calculating the area under the curve of the PDF over a specified interval. This requires integration.

📐 PDF Properties and Probability Calculation

For a continuous random variable $X$ :

$f_X(x) \geq 0$ for all $x$
$\int_{-\infty}^{\infty} f_X(x) \, dx = 1$
The probability that $X$ falls within an interval $[a, b]$ is given by: $P(a \leq X \leq b) = \int_{a}^{b} f_X(x) \, dx$

pythonEvaluating a Normal Distribution PDF

Standard Normal Distribution PDF with Probability Interval

This line chart displays the Probability Density Function (PDF) of a standard normal distribution (mean = 0, standard deviation = 1). The x-axis represents the continuous variable, and the y-axis shows the probability density. The shaded area between -1 and 1 on the x-axis represents the probability

P(-1 \leq X \leq 1)

, which is approximately 0.6827. This visually demonstrates that probability for continuous variables is the area under the curve.

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Key Insight: The area under the PDF curve over an interval represents the probability of the variable falling within that interval.

Check Your Understanding

For a continuous random variable, why is

P(X=c) = 0

for any specific value

c

Comparing PMF and PDF

While both PMF and PDF serve to describe the distribution of a random variable, their fundamental differences stem from the nature of the data they represent. Understanding these distinctions is critical for correctly interpreting probabilities and applying statistical methods. The key lies in how probability is assigned and aggregated across the variable's range.

PMF vs. PDF: Key Distinctions

Feature	Probability Mass Function (PMF)	Probability Density Function (PDF)
Variable Type	Discrete (countable outcomes)	Continuous (uncountable outcomes within a range)
Output Value	Direct probability $P(X=x)$	Probability density $f_X(x)$ (not a direct probability)
Value Range	$0 \leq f_X(x) \leq 1$	$f_X(x) \geq 0$ (can be $>1$ )
Total Probability	Sums to 1: $\sum f_X(x) = 1$	Integrates to 1: $\int f_X(x) \, dx = 1$
Probability for Single Point	Non-zero probability $P(X=x) > 0$	Zero probability $P(X=x) = 0$
Graphical Representation	Bar chart (heights are probabilities)	Smooth curve (area under curve is probability)

A side-by-side comparison highlighting the core differences between PMF and PDF.

pythonUsing `scipy.stats` for PMF and PDF

Try It Yourself

Modify the binomial distribution example to calculate the PMF for 5 trials with a success probability of 0.7. Then, find the probability of exactly 3 successes.

python

Key Takeaways

PMF for Discrete Variables: A Probability Mass Function assigns a direct probability to each specific, countable outcome of a discrete random variable. The sum of all PMF values must equal 1.
PDF for Continuous Variables: A Probability Density Function describes the relative likelihood of outcomes for a continuous random variable. Its value at a point is not a probability; it represents density.
Probability from PDF: For continuous variables, probability is found by calculating the area under the PDF curve over a given interval, typically using integration or a Cumulative Distribution Function (CDF).
Single Point Probability: For discrete variables, $P(X=x) > 0$ . For continuous variables, $P(X=x) = 0$ for any single point $x$ , because a point has no width.
Graphical Interpretation: PMFs are visualized with bar charts where bar heights are probabilities. PDFs are visualized with smooth curves, and probabilities are represented by areas under the curve.
Practical Use: Libraries like scipy.stats provide functions (.pmf() and .pdf()) to easily work with these distributions for both discrete and continuous data analysis.

← All lessons in Data Distributions

Probability Density/Mass Functions (PDF/PMF)

Probability Mass Function (PMF) for Discrete Variables

Probability Density Function (PDF) for Continuous Variables

Comparing PMF and PDF

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