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

Les distributions de probabilité continues modélisent des variables aléatoires pouvant prendre n’importe quelle valeur réelle dans une plage. Par exemple, la taille des femelles adultes peut être de 163,5, 165,25 centimètres, ou n’importe quelle valeur intermédiaire. Cela fait de la hauteur une variable aléatoire continue.La probabilité de cette variable est décrite à l’aide d’une fonction densité de probabilité — une courbe lisse sur l’étendue de la variable. L’unité de cette densité est l’inverse de l’unité de la variable.La probabilité totale que la variable se situe dans un intervalle spécifique est déterminée en calculant la surface sous la courbe pour cette plage. Par exemple, la probabilité que la taille d’une femme soit comprise entre 150 et 170 centimètres est déterminée en intégrant la fonction de densité sur cette plage.Si le résultat est de 0,75, cela signifie que 75 % des femmes de la population ont une taille dans cette période.Mais pour une hauteur exacte, la probabilité est nulle car l’intégration sur un seul point donne une surface nulle.Une fonction de densité de probabilité valide est toujours non négative et s’intègre à 1 sur toute la plage des valeurs possibles pour la variable aléatoire.

applications de l’intégration aux fonctions de densité de probabilité

Applications of Integration to Probability Density Functions

Les distributions de probabilit&#xE9; continues sont utilis&#xE9;es pour mod&#xE9;liser les variables al&#xE9;atoires pouvant prendre n&#x2019;importe quelle valeur r&#xE9;elle dans un intervalle donn&#xE9;. Ces variables ne prennent pas de valeurs isol&#xE9;es ou d&#xE9;nombrables, mais se r&#xE9;partissent sur un continuum. Par exemple, la taille d&#x2019;un individu peut &#xEA;tre mesur&#xE9;e avec une pr&#xE9;cision croissante (par exemple, 163,5 cm ou 165,25 cm), ce qui montre que la taille est une variable al&#xE9;atoire continue.Le comportement de telles variables est d&#xE9;crit par une fonction de densit&#xE9; de probabilit&#xE9;, ou FDP, qui d&#xE9;finit la distribution des probabilit&#xE9;s sur l&#x2019;ensemble des valeurs possibles. Contrairement aux distributions discr&#xE8;tes, o&#xF9; chaque issue a une probabilit&#xE9; d&#xE9;termin&#xE9;e, les distributions continues attribuent des probabilit&#xE9;s &#xE0; des intervalles de valeurs. La probabilit&#xE9; totale qu&#x2019;une variable prenne des valeurs dans un intervalle donn&#xE9; est repr&#xE9;sent&#xE9;e par l&#x2019;aire sous la courbe de la FDP sur cet intervalle. Ce calcul s&#x2019;effectue par int&#xE9;gration :\begin{equation*}P(a \leq X \leq b) = \int_{a}^{b} f(x)\, dx\end{equation*}Par exemple, si l&#x2019;aire sous la courbe entre 150 et 170 cm est de 0,75, il y a 75 % de chances qu&#x2019;une personne choisie au hasard ait une taille comprise dans cet intervalle. Cependant, la probabilit&#xE9; que la variable prenne une valeur exacte est toujours nulle, car l&#x2019;aire associ&#xE9;e &#xE0; un point isol&#xE9; est nulle.Pour &#xEA;tre valide, une fonction de densit&#xE9; de probabilit&#xE9; doit &#xEA;tre non n&#xE9;gative pour toutes les valeurs et son int&#xE9;grale doit &#xEA;tre &#xE9;gale &#xE0; 1 sur tout le domaine de la variable. Ces propri&#xE9;t&#xE9;s garantissent que la fonction repr&#xE9;sente fid&#xE8;lement une distribution de probabilit&#xE9; et que la probabilit&#xE9; totale sur l&#x2019;ensemble des valeurs possibles est &#xE9;gale &#xE0; 1.

Applications de l’intégration aux fonctions de densité de probabilité

Les distributions de probabilit&#xE9; continues sont utilis&#xE9;es pour mod&#xE9;liser les variables al&#xE9;atoires pouvant prendre n&#x2019;importe ...

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

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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

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.

Finding Probability with Area Under a PDF | JoVE K12

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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