Probability Density Function of a Continuous Random Variable

A continuous random variable cannot be described by listing probabilities for individual values, because $P(X = x) = 0$ for every single $x$ — there are simply too many (uncountably many) values in any interval for any one of them to carry a positive probability. Instead, a continuous random variable is described by a probability density function, written $f(x)$, and probabilities are read off as the area under the curve of $f(x)$ over an interval, rather than the height of the curve at a single point.

Conditions for a valid p.d.f.

A function $f(x)$ is a valid probability density function over a range if:

The second condition is simply the continuous counterpart of "the probabilities must add up to $1$" — instead of a sum, we use an integral because $x$ varies continuously.

For any two numbers $a \leq b$ in the range of $X$, the probability that $X$ falls between them is:

This is exactly the area under the curve $y = f(x)$ between $x = a$ and $x = b$. Because a single point has zero width, $P(X = a) = 0$, and so for a continuous random variable it makes no difference whether an inequality is strict or not: $P(X \leq a) = P(X < a)$.

Illustration. Consider $f(x) = 2x$ for $0 \leq x \leq 1$, and $f(x) = 0$ elsewhere. Since $f(x) = 2x \geq 0$ throughout $[0,1]$, and

$\int_{0}^{1} 2x\, dx = \big[x^2\big]_{0}^{1} = 1 - 0 = 1,$ …