The probability density function (pdf) for a continuous [[random variable]] defines the curve under which area represents probability, usually denoted $f(x)$. The corollary for a discrete random variable is the [[probability mass function]]. For any continuous random variable, the probability for any specific value is always $0$. For continuous random variables we instead think about the probability of finding a value in a range, or To find this probability, we integrate the pdf. $P(a <= X <= b) = \int_a^bf(x)dx$ # expanded notation When a pdf is parameterized by some [[parameter]] $\theta$, denote the pdf as follows to emphasize the dependence of the pdf on $\theta$. $f(x; \theta)$ Separate multiple parameters by commas. For a [[joint probability density function]], separate multiple random variables by commas. ## properties The probability at any given $x$ must be greater than 0. $f(x) >0$ The integral from $-\infty$ to $\infty$ a for a pdf must equal 1. You can use this to confirm that any derived pdf is valid. $\int\limits_{-\infty}^{\infty}f(x)dx = 1$ [[kernel]]