probability density function

(noun)

any function whose integral over a set gives the probability that a random variable has a value in that set

Examples of probability density function in the following topics:

  • Probability

    • Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.
    • Here, we will learn what probability distribution function is and how it functions with regard to integration.
    • In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
    • The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.
    • A probability density function is most commonly associated with absolutely continuous univariate distributions.

  • Applications of Multiple Integrals

    • As is the case with one variable, one can use the multiple integral to find the average of a function over a given set.
    • Given a set $D \subseteq R^n$ and an integrable function $f$ over $D$, the average value of $f$ over its domain is given by:
    • In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of the density weighed with the square of the distance from the axis:
    • If there is a continuous function $\rho(x)$ representing the density of the distribution at $x$, so that $dm(x) = \rho (x)d^3x$, where $d^3x$ is the Euclidean volume element, then the gravitational potential is:
    • In the following example, the electric field produced by a distribution of charges given by the volume charge density $\rho (\vec r)$ is obtained by a triple integral of a vector function:

  • Indefinite Integrals and the Net Change Theorem

    • $f(x)$, the function being integrated, is known as the integrand.
    • Note that the indefinite integral yields a family of functions.
    • For example, the function $F(x) = \frac{x^3}{3}$ is an antiderivative of $f(x) = x^2$.
    • Essentially, the graphs of antiderivatives of a given function are vertical translations of each other, with each graph's location depending upon the value of $C$.
    • This can be applied to find values such as volume, concentration, density, population, cost, and velocity.