Interactive PDF Calculator
Choose a distribution, enter parameters, and calculate the probability density at a point x.
Tip: PDF values can be greater than 1. They are densities, not direct probabilities.
What this probability density function calculator does
This tool computes the value of a probability density function (PDF) at a specific point. PDFs are used with continuous random variables such as heights, waiting times, lifetimes, signal noise, and financial returns. Instead of asking for the probability of one exact value (which is zero in continuous models), we use the PDF to understand how concentrated probability is around that value.
Supported distributions and formulas
The calculator currently supports four commonly used continuous distributions:
| Distribution | Support | |
|---|---|---|
| Normal (μ, σ) | f(x) = 1 / (σ√(2π)) · exp(-(x-μ)2 / (2σ2)) | x ∈ ℝ, σ > 0 |
| Exponential (λ) | f(x) = λe-λx | x ≥ 0, λ > 0 |
| Uniform (a, b) | f(x) = 1/(b-a) for a ≤ x ≤ b, else 0 | a < b |
| Gamma (k, θ) | f(x) = xk-1e-x/θ / (Γ(k)θk) | x ≥ 0, k > 0, θ > 0 |
How to use the calculator
- Select your distribution from the dropdown.
- Enter valid distribution parameters (for example, σ must be positive for a normal distribution).
- Enter the point x where you want the density.
- Click Calculate PDF to view the result and the exact formula used.
How to interpret the output
1) A density is not a direct probability
For continuous variables, probabilities come from area under the curve, not from a single PDF value. In other words, the probability of landing in an interval [x1, x2] is the integral of f(x) over that interval.
2) A higher PDF means “more locally likely”
If f(2.1) is larger than f(5.7), values near 2.1 are more concentrated in the distribution than values near 5.7. This helps with comparing relative likelihoods across locations.
Worked examples
Normal example
Set μ = 0, σ = 1, and x = 0. You should get f(0) ≈ 0.398942, the well-known peak of the standard normal curve.
Exponential example
Set λ = 0.5 and x = 3. The result is f(3) = 0.5e-1.5 ≈ 0.111565. This is a common model for waiting-time processes with constant hazard.
Uniform example
Set a = 10 and b = 20. Any x between 10 and 20 returns the same density 1/10 = 0.1. Values outside [10, 20] return 0.
Common mistakes to avoid
- Using non-positive standard deviation for normal distributions.
- Using λ ≤ 0 for exponential distributions.
- Setting b ≤ a for uniform distributions.
- Treating PDF(x) as P(X = x), which is not correct for continuous variables.
Why this matters in practice
PDF calculations appear in machine learning, Bayesian statistics, reliability engineering, economics, and scientific modeling. Whether you are fitting a model, checking assumptions, or building intuition about uncertainty, quickly evaluating density values is an essential skill.