Mathematics
Probability Density Function
Quick fact
A probability density function can take values greater than 1 — for example, a uniform distribution on the interval [0, 0.5] has a constant density of 2.
Why this is interesting
If you roll a fair six-sided die, each face has a probability of 1/6. But what if you spin a wheel that can stop at any angle from 0 to 360 degrees? How can we talk about the probability of landing at exactly 90.000... degrees?
Read the full explanation
Understanding Probability Density Function
Imagine a spinner that can stop at any real number between 0 and 1. Since there are infinitely many possible values, the chance of landing on any specific number (like exactly 0.5) is zero. Instead, we talk about the probability that the result lies within a range, such as between 0.4 and 0.6. The probability density function is a curve that tells us how 'dense' the probability is at each point. The higher the curve in a region, the more likely the value falls in that region. To find the actual probability for an interval, we compute the area under the curve over that interval. The total area under the entire curve must be 1, representing 100% probability.
A deeper explanation
The probability density function (PDF) is a mathematical function f(x) that describes the relative likelihood of a continuous random variable X. The key rule: the probability that X lies between a and b equals the integral of f(x) from a to b. Because individual points have zero width, their probability is zero — we only consider intervals. The height f(x) is not a probability; it's a density, measured in probability per unit of x. For the PDF to be valid, it must be non-negative for all x, and its integral over the entire real line must equal 1. This normalization ensures the total probability is unity. The PDF is intimately linked to the cumulative distribution function (CDF): the CDF F(x) gives P(X ≤ x), and the PDF is the derivative of the CDF. Understanding the PDF allows us to compute expected values, variances, and other moments of continuous distributions, making it indispensable for modeling everything from physical measurements to financial risk.