A probability mass function (PMF) gives the probability of each possible value of a discrete random variable. A probability density function (PDF) describes a continuous random variable; to find a probability from it, integrate the density over a range. A density value at one point is not the probability of that point. The cumulative distribution function (CDF) provides a common way to express probabilities for either kind of variable.
PMF vs. PDF: the key difference
| Question | Probability mass function (PMF) | Probability density function (PDF) |
|---|---|---|
| Used for | A discrete random variable with finite or countable possible values | A continuous random variable represented by a density |
| What the function gives | For a supported value x, p(x) = P(X = x) | The density at x, not P(X = x) |
| How to calculate an event probability | Add the masses for the values in the event | Integrate the density over the event’s interval or region |
| How it is normalized | The masses sum to 1 | The density integrates to 1 over its domain |
| Probability at one exact value | Can be positive for a supported value | Is zero for an individual point when the variable has a continuous density |
| Example | A die roll | A measured lifetime, distance, or weight |
These definitions follow Boston University’s probability and statistics refresher, Penn State STAT 500, and Penn State STAT 414.
What a PMF tells you
For a discrete random variable X, its probability mass function is defined by p(x) = P(X = x). It assigns a probability to each possible value in the variable’s support. A valid PMF is nonnegative, assigns zero probability outside the support, and has masses that sum to 1.
Example: one fair die roll
Let X be the number of spots on one roll of a fair six-sided die. The possible values are 1, 2, 3, 4, 5, and 6, each with probability 1/6. To find P(X ≤ 2), add the masses for 1 and 2: 1/6 + 1/6 = 1/3.
#1 Best Overall
- Read Before You Buy — No Video Output: These adapters support charging and USB 2.0 data transfer, but cannot transmit video signals. Except for standard USB webcams (which use USB data only), they are not compatible with HDMI/DisplayPort cables, video-capable USB-C hubs, or docking stations with video output.
- Convert USB-A Ports to USB-C: Designed to connect USB-C earphones, cables, flash drives, card readers, and other USB-C accessories to standard USB-A ports. Plug-and-play with no drivers or software required.
- Aluminum Alloy Housing: Built with a sturdy aluminum alloy shell that aids in heat dissipation and protects against daily wear and scratches. Designed to maintain a stable and secure connection.
- Compact & Travel-Friendly: The ultra-compact design allows the adapter to stay plugged into your device without blocking adjacent ports or adding bulk, reducing wear and tear on your original USB ports.
- 12-Month Warranty: Backed by a 12-month manufacturer warranty for peace of mind. Designed to meet strict quality control standards for reliable everyday performance.
More generally, for a set of values A, calculate P(X ∈ A) by summing p(x) across the values x in A. The notation for a PMF can vary: some sources use p(x), while others use f(x). What matters is that, in the discrete case, the function’s value is the probability of the corresponding outcome.
What a PDF tells you—and why its height is not a probability
A probability density function describes a continuous random variable. A valid density is nonnegative and integrates to 1 over its domain. Its value f(x) is a density, not the probability that X equals x. For a variable with a continuous density, P(X = x) = 0 for any single value x.
Rank #2
- 5-in-1 USB-C Hub: Experience comprehensive connectivity featuring a Power Delivery input, two USB-A 2.0 ports, a USB-A 3.0 port, and an HDMI port. (Note: The USB-C power delivery input port is only for connecting an external wall charger to power your laptop and cannot power peripheral devices.)
- 90W Pass-Through Charging: Achieve optimal charging with 90W pass-through power to your laptop, supported by a total input of 100W, with the hub reserving 10W for operational efficiency. (Note: Wall charger not included.)
- Quick Data Transfers: Accelerate your productivity with rapid data transfers using a high-speed 5Gbps USB 3.0 port and two 480Mbps USB 2.0 ports.
- 4K HDMI Display: Enhance your visual experience with a hub capable of delivering 4K resolution at 30Hz in both mirror and extend modes. Please note that this hub is compatible with MacBook (macOS 12 and newer), Windows 10 and 11, ChromeOS, and laptops equipped with DP Alt Mode and Power Delivery. Note: This device is not compatible with Linux.
- What You Get: Anker USB-C Hub (5-in-1, 4K HDMI), welcome guide, 18-month warranty, and our friendly customer service.
To find the probability that X falls between a and b, integrate the density across that interval: P(a ≤ X ≤ b) = ∫ab f(x) dx. Geometrically, this integral is the area under the density curve over the interval. A density height can even exceed 1; that does not violate the probability rules, because probability is determined by area, not height alone.
Example: a measurement
Suppose X represents a randomly selected hamburger’s weight. A question such as “What is the chance that X is between 0.20 and 0.30 pounds?” asks for the probability across an interval, calculated by integrating the density from 0.20 to 0.30. The exact weight of 0.25 pounds is a single point, so it has probability zero under a continuous-density model.
Free tools Windows power users keep installed
One-click scans. No signup required.
Rank #3
- Sleek 7-in-1 USB-C Hub: Features an HDMI port, two USB-A 3.0 ports, and a USB-C data port, each providing 5Gbps transfer speeds. It also includes a USB-C PD input port for charging up to 100W and dual SD and TF card slots, all in a compact design.
- Flawless 4K@60Hz Video with HDMI: Delivers exceptional clarity and smoothness with its 4K@60Hz HDMI port, making it ideal for high-definition presentations and entertainment. (Note: Only the HDMI port supports video projection; the USB-C port is for data transfer only.)
- Double Up on Efficiency: The two USB-A 3.0 ports and a USB-C port support a fast 5Gbps data rate, significantly boosting your transfer speeds and improving productivity.
- Fast and Reliable 85W Charging: Offers high-capacity, speedy charging for laptops up to 85W, so you spend less time tethered to an outlet and more time being productive.
- What You Get: Anker USB-C Hub (7-in-1), welcome guide, 18-month warranty, and our friendly customer service.
The same reasoning applies to continuous models of quantities such as lifetime or distance. Measurement values may be recorded to a particular precision, but that does not make the underlying continuous model a PMF over every recorded decimal.
How the CDF connects both cases
The cumulative distribution function (CDF) is F(x) = P(X ≤ x). It gives the probability that the variable is at or below a threshold.
Rank #4
- Dual Converters, Infinite Potential：Includes 2× USB C male to USB A female adapters and 2× USB A male to USB C female adapters. Perfect for a wide range of uses—tablets with Bluetooth keyboards, expand USB ports on macbook, and more. Two different converters for all your daily needs
- Next-Level 10Gbps & 3A Charging: No more slow 480Mbps, this usb to usb c adapter has a transfer speed of up to 10Gbps, allowing you to do more transferring in less time. This usb adapter fits both USB A and USB C charger, supporting up to 3A fast charging
- Upgraded Exquisite Craftsmanship: With an aluminum alloy housing and metal connector, the usbc to usb adapter is extremely durable and sturdy. Rigorously tested to withstand more than 10,000 times of plugging and unplugging, ensuring long-lasting performance
- Broad Compatible: The usb c to usb adapter widely supports all USB C/ USB A devices like laptops, tablets, cellphones, car chargers, and phone chargers. Such as compatible with MacBook Pro/Air 2023/2022, Thunderbolt 4/3 Devices,Apple MagSafe Watch 9/8/7/SE/Ultra, iPad Pro 2022/2021, Samsung Galaxy S23/S20/S10, and iPhone 17/16/15 Pro. Plug and play
- Please Note: To reach 10Gbps speed, keep the cable under 3.3 ft. For USB A Male to USB C adapters, try flipping the USB C connector. USB C Male to USB A adapters support bidirectional 10Gbps transfer within 3.3 ft
- For a discrete variable, the CDF adds the PMF values up through x.
- For a continuous variable with a PDF, the CDF integrates the density up through x.
Where the CDF is differentiable, its derivative is the PDF. The CDF therefore offers a shared cumulative view, although the PMF and PDF themselves answer different kinds of probability questions.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How to choose the right function
- Identify what X represents. If its possible values are countable outcomes, such as the spots on a die, use the discrete framework. If it measures a quantity such as weight or distance, a continuous framework may be appropriate. OpenStax explains this distinction through counting versus measuring, and Penn State STAT 414 discusses countable and uncountable supports.
- Ask whether the question is about an exact value or a range. For a discrete variable, a PMF can give the probability of an exact supported value. For a continuous variable with a density, find the probability over an interval by integration.
- Check the normalization. Confirm that the PMF’s probabilities sum to 1 or that the PDF integrates to 1 over the full domain.
- Use the CDF for cumulative questions. For questions phrased as “at most x,” calculate or read F(x) = P(X ≤ x), using a cumulative sum for a PMF or a cumulative integral for a PDF.
Clarifying the term “probability distribution function”
“Probability distribution function” is not precise enough by itself to identify which function is meant. Specify whether you mean a probability mass function (PMF), a probability density function (PDF), or a cumulative distribution function (CDF). In this article, PDF means probability density function, not the document-file format.
What’s actually slowing this PC down?
Pick the symptom - the matching free tool is one click away.
Best Value
- 5-in-1 Connectivity: Equipped with a 4K HDMI port, a 5 Gbps USB-C data port, two 5 Gbps USB-A ports, and a USB C 100W PD-IN port. Note: The USB C 100W PD-IN port supports only charging and does not support data transfer devices such as headphones or speakers.
- Powerful Pass-Through Charging: Supports up to 85W pass-through charging so you can power up your laptop while you use the hub. Note: Pass-through charging requires a charger (not included). Note: To achieve full power for iPad, we recommend using a 45W wall charger.
- Transfer Files in Seconds: Move files to and from your laptop at speeds of up to 5 Gbps via the USB-C and USB-A data ports. Note: The USB C 5Gbps Data port does not support video output.
- HD Display: Connect to the HDMI port to stream or mirror content to an external monitor in resolutions of up to 4K@30Hz. Note: The USB-C ports do not support video output.
- What You Get: Anker 332 USB-C Hub (5-in-1), welcome guide, our worry-free 18-month warranty, and friendly customer service.
The PMF-versus-PDF comparison describes the standard discrete and continuous cases. It is a useful first distinction, not a claim that every possible probability distribution must fit one of those two elementary forms.
Quick Recap
Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.
