A continuous outcome never hits one exact number
Ask for the chance a bacterium dies at exactly five hours and the answer is zero. Continuous chance lives in intervals, not pinpoints. A probability density function turns height into relative likelihood per unit, and only the area underneath becomes probability.
For an absolutely continuous random variable, a probability density function assigns a relative likelihood at each point in the sample space. The absolute chance of landing on any single precise value is zero. Comparing densities at two points says how much more likely a draw is to fall near one than the other. Probability for a range is the integral of the density over that range—the nonnegative area under the curve between the ends.
The density stays nonnegative everywhere, and the total area under the whole curve equals one, so something in the possible set is certain. A bacterium that typically lives twenty to thirty hours has zero chance of dying at exactly five hours measured to infinite precision, yet dying between five and 5.01 hours might have probability 0.02. Shrinking the window by ten shrinks that chance by about ten, so the ratio of probability to duration stays near two per hour—the density at five hours.
That density can exceed one: a continuous uniform law on the interval from zero to one half has density two there and zero elsewhere. The standard normal density is one over square root of two pi times e to the minus x squared over two. Expectation, when it exists, is the integral of x times f(x). Discrete laws use mass functions instead; the Cantor distribution also lacks a density despite putting no mass on single points.
A distribution admits a density when its cumulative distribution function is absolutely continuous; then the CDF's derivative is essentially the density. Two density formulas name the same law whenever they disagree only on a Lebesgue-null set. Dirac deltas can encode some discrete laws as generalized densities, as with a Rademacher variable that equals minus one or one with equal chance one half.
Source: Probability density function
