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

Distribuições de probabilidade contínuas modelam variáveis aleatórias que podem assumir qualquer valor real dentro de um intervalo. Por exemplo, a altura das fêmeas adultas pode ser 163,5, 165,25 centímetros ou qualquer valor intermediário. Isso faz da altura uma variável aleatória contínua.A probabilidade dessa variável é descrita usando uma função densidade de probabilidade — uma curva suave sobre o intervalo da variável. A unidade dessa densidade é o recíproco da unidade da variável.A probabilidade total de que a variável esteja dentro de um intervalo específico é calculada calculando a área sob a curva para esse intervalo. Por exemplo, a probabilidade de a altura de uma mulher estar entre 150 e 170 centímetros é encontrada integrando a função densidade nessa faixa.Se o resultado for 0,75, significa que 75% das mulheres na população têm alturas dentro desse intervalo.Mas para uma altura exata, a probabilidade é zero porque integrar sobre um único ponto dá área zero.Uma função densidade de probabilidade válida é sempre não negativa e integra a 1 em toda a faixa de valores possíveis para a variável aleatória.

aplicações da integração em funções de densidade de probabilidade

Applications of Integration to Probability Density Functions

Distribui&#xE7;&#xF5;es de probabilidade cont&#xED;nuas s&#xE3;o utilizadas para modelar vari&#xE1;veis aleat&#xF3;rias que podem assumir qualquer valor real dentro de um intervalo determinado. Essas vari&#xE1;veis n&#xE3;o assumem valores isolados ou enumer&#xE1;veis, mas existem em um cont&#xED;nuo. Por exemplo, a altura de um indiv&#xED;duo pode ser medida com precis&#xE3;o crescente, como 163,5 ou 165,25 cm, demonstrando que a altura &#xE9; uma vari&#xE1;vel aleat&#xF3;ria cont&#xED;nua.O comportamento de tais vari&#xE1;veis &#xE9; descrito por meio de uma fun&#xE7;&#xE3;o de densidade de probabilidade (PDF), que define como as probabilidades s&#xE3;o distribu&#xED;das no intervalo de valores poss&#xED;veis. Ao contr&#xE1;rio das distribui&#xE7;&#xF5;es discretas, em que resultados individuais t&#xEA;m probabilidades espec&#xED;ficas, as distribui&#xE7;&#xF5;es cont&#xED;nuas atribuem probabilidades a intervalos de valores. A probabilidade total de uma vari&#xE1;vel cair em um intervalo espec&#xED;fico &#xE9; representada pela &#xE1;rea sob a curva da PDF nesse intervalo. Isso &#xE9; calculado por meio de integra&#xE7;&#xE3;o:\begin{equation*}P(a \leq X \leq b) = \int_{a}^{b} f(x)\, dx\end{equation*}Por exemplo, se a &#xE1;rea sob a curva entre 150 e 170 cm for 0,75, ent&#xE3;o h&#xE1; 75 % de probabilidade de que um indiv&#xED;duo selecionado aleatoriamente tenha uma altura nesse intervalo. No entanto, a probabilidade de a vari&#xE1;vel assumir qualquer valor exato &#xE9; sempre zero, pois a &#xE1;rea sob um &#xFA;nico ponto &#xE9; zero.Para ser v&#xE1;lida, uma fun&#xE7;&#xE3;o de densidade de probabilidade deve ser n&#xE3;o negativa para todos os valores e deve integrar a 1 em todo o dom&#xED;nio da vari&#xE1;vel. Essas propriedades garantem que a fun&#xE7;&#xE3;o represente fielmente uma distribui&#xE7;&#xE3;o de probabilidade e que a probabilidade total associada a todos os resultados poss&#xED;veis seja igual a 1.

Aplicações da integração em funções de densidade de probabilidade

Distribui&#xE7;&#xF5;es de probabilidade cont&#xED;nuas s&#xE3;o utilizadas para modelar vari&#xE1;veis aleat&#xF3;rias que podem assumir qualquer valor ...

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

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Calculus

Cálculo

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Application of Techniques of Integration

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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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