Name: finding probability with area under a pdf
Uploaded: 2026-01-12T12:24:35.000Z
Duration: 1 min 27 s
Channel: JoVE

Kontinuierliche Wahrscheinlichkeitsverteilungen modellieren Zufallsvariablen, die innerhalb eines Bereichs jeden realen Wert annehmen können. Zum Beispiel könnte die Körpergröße erwachsener Frauen 163,5, 165,25 Zentimeter oder ein beliebiger Wert dazwischen betragen. Dadurch ist die Höhe eine kontinuierliche Zufallsvariable.Die Wahrscheinlichkeit dieser Variablen wird mit einer Wahrscheinlichkeitsdichtefunktion beschrieben – einer glatten Kurve über den Bereich der Variablen. Die Einheit dieser Dichte ist der Reziproke der Einheit der Variablen.Die Gesamtwahrscheinlichkeit, dass die Variable innerhalb eines bestimmten Intervalls liegt, wird durch Berechnung der Fläche unter der Kurve für diesen Bereich ermittelt. Zum Beispiel wird die Wahrscheinlichkeit, dass die Körpergröße einer Frau zwischen 150 und 170 Zentimetern liegt, durch Integration der Dichtefunktion über diesen Bereich ermittelt.Wenn das Ergebnis 0,75 ist, bedeutet das, dass 75 % der Frauen in der Bevölkerung innerhalb dieses Intervalls eine Körpergröße haben.Aber für eine exakte Höhe ist die Wahrscheinlichkeit null, weil Integration über einen einzelnen Punkt eine Fläche von null ergibt.Eine gültige Wahrscheinlichkeitsdichtefunktion ist immer nicht-negativ und integriert sich über den gesamten Bereich der möglichen Werte für die Zufallsvariable zu 1.

anwendungen der integration auf wahrscheinlichkeitsdichtefunktionen

Applications of Integration to Probability Density Functions

Stetige Wahrscheinlichkeitsverteilungen werden verwendet, um Zufallsvariablen zu modellieren, die jeden reellen Wert innerhalb eines bestimmten Bereichs annehmen k&#xF6;nnen. Diese Variablen nehmen keine isolierten oder abz&#xE4;hlbaren Werte an, sondern existieren auf einem stetigen Kontinuum. Beispielsweise kann die K&#xF6;rpergr&#xF6;&#xDF;e einer Person mit zunehmender Genauigkeit gemessen werden &#x2013; etwa 163,5 cm oder 165,25 cm &#x2013;, was verdeutlicht, dass es sich bei der K&#xF6;rpergr&#xF6;&#xDF;e um eine stetige Zufallsvariable handelt.Das Verhalten solcher Variablen wird durch eine Wahrscheinlichkeitsdichtefunktion beschrieben, die darlegt, wie Wahrscheinlichkeiten &#xFC;ber den Bereich m&#xF6;glicher Werte verteilt sind. Im Gegensatz zu diskreten Verteilungen, bei denen einzelnen Ergebnissen spezifische Wahrscheinlichkeiten zugeordnet werden, weisen stetige Verteilungen Wahrscheinlichkeiten Werteintervallen zu. Die Gesamtwahrscheinlichkeit, dass eine Variable in ein bestimmtes Intervall f&#xE4;llt, wird durch die Fl&#xE4;che unter der Wahrscheinlichkeitsdichtefunktionskurve &#xFC;ber diesem Bereich dargestellt und mittels Integration berechnet:\begin{equation*}P(a \leq X \leq b) = \int_{a}^{b} f(x)\, dx\end{equation*}Beispielsweise gilt: Wenn die Fl&#xE4;che unter der Kurve zwischen 150 und 170 cm 0,75 betr&#xE4;gt, dann liegt die Wahrscheinlichkeit, dass eine zuf&#xE4;llig ausgew&#xE4;hlte Person eine K&#xF6;rpergr&#xF6;&#xDF;e innerhalb dieses Intervalls aufweist, bei 75 %. Die Wahrscheinlichkeit, dass die Variable einen exakt bestimmten Wert annimmt, ist hingegen stets null, da die Fl&#xE4;che unter einem einzelnen Punkt null ist.Eine Wahrscheinlichkeitsdichtefunktion muss f&#xFC;r alle Werte nichtnegativ sein und &#xFC;ber den gesamten Definitionsbereich der Variablen zu 1 integrieren, um g&#xFC;ltig zu sein. Diese Eigenschaften stellen sicher, dass die Funktion eine Wahrscheinlichkeitsverteilung korrekt repr&#xE4;sentiert und die Gesamtwahrscheinlichkeit &#xFC;ber alle m&#xF6;glichen Ergebnisse 1 betr&#xE4;gt.

Anwendungen der Integration auf Wahrscheinlichkeitsdichtefunktionen

Stetige Wahrscheinlichkeitsverteilungen werden verwendet, um Zufallsvariablen zu modellieren, die jeden reellen Wert innerhalb eines bestimmten Bereichs ...

applications-of-integration-to-probability-density-functions

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JoVE Core

Calculus

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Application of Techniques of Integration

Anwendung von Integrationstechniken

KEY TERMS AND CONCEPTS

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finding probability with area under a pdf

Continuous probability distributions describe random variables that can take any real value within a set range. Height is one example, since it can be measured with growing precision, such as 163.5 or 165.25 centimeters. That makes height a continuous random variable.

The pattern of a continuous random variable is shown with a probability density function, or PDF. A PDF shows how probability is spread across the possible values. In a continuous distribution, probabilities are assigned to intervals of values, not to single outcomes.

Integration gives the total probability for a chosen interval. The area under the PDF curve over that interval is the probability. For example, if the area under the curve from 150 to 170 centimeters is 0.75, then there is a 75% chance that a randomly selected individual falls in that height range.

A single exact value has probability zero in a continuous distribution. That is because the area under one point is zero. A valid PDF must also stay non-negative for all values and integrate to 1 over the entire domain. These rules make sure the function represents a true probability distribution and that the total probability across all possible outcomes equals 1.

Finding Probability with Area Under a PDF

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applications of integration to probability density functions

Continuous probability distributions describe random variables that can take any real value within a set range. Height is one example, since it can be measured with growing precision, such as 163.5 or 165.25 centimeters. That makes height a continuous random variable. The pattern of a continuous random variable is shown with a probability density function, or PDF. A PDF shows how probability is spread across the possible values. In a continuous distribution, probabilities are assigned to...

key terms and concepts

Learn how continuous probability distributions use PDFs and integration to find probability from the area under a curve, using height as an example.

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The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Probability with Area Under a PDF

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