Density curve

A density curve is a curve $f(x)$ with exactly two requirements. It never drops below the horizontal axis, so $f(x) \ge 0$ everywhere, and the total area between the curve and the axis is exactly 1. Area is the only quantity you read off it: the area above an interval from $a$ to $b$ is the proportion of the distribution lying between them, and for a value drawn at random it is $P(a < X < b)$.

Spread a distribution uniformly over 0 to 0.4. The curve is a flat rectangle whose height is forced by the total-area rule: $1 / 0.4 = 2.5$, since $0.4 \times 2.5 = 1$. The area from 0.1 to 0.3 is then $0.2 \times 2.5 = 0.50$, so exactly half the distribution sits in that stretch.

That height is where the usual mistake lives. "The curve is 2.5 at $x = 0.2$, so a value there has probability 2.5" is wrong twice over. A density value is not a probability. It is a rate, probability per unit along the horizontal axis, with no ceiling: squeeze the same total area of 1 into a narrower interval and the height climbs as far as you like. Only a height multiplied by a width is a probability. The same arithmetic settles the second half. A single point has no width, so it encloses no area, so $P(X = c) = 0$ for any one value $c$, which is why $P(X < c)$ and $P(X \le c)$ agree on a density curve and do not agree for counts.

Heights are still worth comparing to each other. A curve twice as tall at one place as at another says values are packed twice as densely there. What a height never is, on its own, is an answer to a question about chance.

A density curve is a model laid over data, not the data. In the Fall 2026 course the idea is anchored in topic 2.11, The Normal Distribution, where the probability of an interval is the area under the curve over it.

Reading a proportion as an area is not an analogy for calculus, it is calculus: the area under $f$ between $a$ and $b$ is the definite integral of the density, and the total-area-equals-1 rule is the statement that the density integrates to 1 over its whole domain.