Explore probability mass functions (PMF) for discrete distributions and probability density functions (PDF) for continuous distributions. Each tool provides interactive parameter controls, real-time chart updates, and detailed explanations with formulas for six discrete and three continuous distributions.
What are Probability Functions?
A probability function assigns probabilities to outcomes of a random variable. The type of function depends on whether the variable is discrete or continuous.
For discrete random variables, the probability mass function (PMF) gives the exact probability that the variable equals each specific value. The PMF satisfies two properties: every probability is between 0 and 1, and all probabilities sum to exactly 1.
For continuous random variables, the probability density function (PDF) describes relative likelihood across a range of values. Unlike the PMF, the PDF value at a point is not a probability—it's a density. Probabilities are found by integrating the PDF over intervals. The total area under any PDF equals 1.
These visualization tools let you explore both types interactively, seeing how parameters affect the shape and spread of each distribution.
Discrete vs Continuous Distributions
The fundamental distinction in probability theory is between discrete and continuous random variables.
Discrete distributions apply to countable outcomes. Examples include: • Number of heads in coin flips (Binomial) • Customer arrivals per hour (Poisson) • Trials until first success (Geometric) • Defects in a sample (Hypergeometric)
The PMF gives exact probabilities like P(X = 5) = 0.246.
Continuous distributions apply to measurements taking any value in an interval. Examples include: • Heights and weights (Normal) • Time until failure (Exponential) • Random positions (Uniform)
The PDF requires integration: P(2 ≤ X ≤ 5) = ∫f(x)dx from 2 to 5.
The discrete tool displays bar charts showing individual probabilities. The continuous tool shows smooth curves with a PDF/CDF toggle to switch between density and cumulative views.
Available Distributions
The visualization tools cover nine fundamental distributions used throughout probability and statistics.
Discrete Distributions (6 types):
• Discrete Uniform: Equal probability for each value in a range • Binomial: Successes in n independent trials • Geometric: Trials until first success • Negative Binomial: Trials until r successes • Hypergeometric: Sampling without replacement • Poisson: Events at constant average rate
Continuous Distributions (3 types):
• Uniform: Constant density over an interval • Normal: Bell-shaped curve, symmetric around mean • Exponential: Time between Poisson events
Each distribution has adjustable parameters so you can explore how changing n, p, λ, μ, or σ affects the shape.
Key Concepts in Probability Functions
Understanding probability functions requires grasping several foundational concepts:
Variance: Measures spread around the mean. Variance equals E[(X-μ)²] and determines distribution width.
Cumulative Distribution Function: The CDF gives P(X ≤ x), showing accumulated probability up to any point. For continuous distributions, the CDF is the integral of the PDF.
Parameters: Each distribution has parameters controlling its shape. Binomial has n and p; Normal has μ and σ; Poisson has λ. The visualizers let you adjust these and see immediate effects.
These concepts connect across all distributions, making it valuable to explore multiple distributions and observe common patterns.
Related Tools and Resources
These probability function visualizers connect to other tools and theory pages on the site: