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

Sürekli olasılık dağılımları, bir aralık içindeki herhangi bir gerçek değeri alabilen rastgele değişkenleri modeller. Örneğin, yetişkin dişilerin boyu 163,5, 165,25 santimetre veya aradaki herhangi bir değer olabilir. Bu da yüksekliği sürekli rastgele değişken yapar.Bu değişkenin olasılığı, değişkenin aralığı üzerinde düzgün bir eğri olan bir olasılık yoğunluk fonksiyonu kullanılarak tanımlanır. Bu yoğunluğun birimi, değişkenin biriminin karşılıklıdır.Değişkenin belirli bir aralık içinde yer alma olasılığı, o aralık için eğrin altındaki alan hesaplanarak hesaplanır. Örneğin, bir kadının boyunun 150 ile 170 santimetre arasında olma olasılığı, yoğunluk fonksiyonunun bu aralık üzerinden entegre edilmesiyle elde edilir.Sonuç 0.75 ise, bu nüfustaki kadınların %75'inin bu aralıkta boylara sahip olduğu anlamına gelir.Ancak tam bir yükseklik için olasılık sıfırdır çünkü tek bir nokta üzerinde entegre olmak sıfır alan verir.Geçerli bir olasılık yoğunluk fonksiyonu her zaman negatif değildir ve rastgele değişken için tüm olası değerler aralığında 1'e entegre olur.

olasılık yoğunluk fonksiyonlarında İntegrasyonun uygulamaları

Applications of Integration to Probability Density Functions

S&#xFC;rekli olas&#x131;l&#x131;k da&#x11F;&#x131;l&#x131;mlar&#x131;, belirli bir aral&#x131;kta herhangi bir ger&#xE7;ek de&#x11F;eri alabilen rastgele de&#x11F;i&#x15F;kenleri modellemek i&#xE7;in kullan&#x131;l&#x131;r. Bu de&#x11F;i&#x15F;kenler, ayr&#x131;k ya da say&#x131;labilir de&#x11F;erler almaz; aksine bir s&#xFC;reklilik &#xFC;zerinde tan&#x131;ml&#x131;d&#x131;r. &#xD6;rne&#x11F;in, bir bireyin boyu artan bir hassasiyetle &#xF6;l&#xE7;&#xFC;lebilir&#x2014;&#xF6;rne&#x11F;in 163,5 ya da 165,25 cm&#x2014;bu durum boyun s&#xFC;rekli bir rastgele de&#x11F;i&#x15F;ken oldu&#x11F;unu g&#xF6;sterir.Bu t&#xFC;r de&#x11F;i&#x15F;kenlerin davran&#x131;&#x15F;&#x131;, olas&#x131;l&#x131;klar&#x131;n m&#xFC;mk&#xFC;n de&#x11F;erler aral&#x131;&#x11F;&#x131; &#xFC;zerinde nas&#x131;l da&#x11F;&#x131;ld&#x131;&#x11F;&#x131;n&#x131; tan&#x131;mlayan bir olas&#x131;l&#x131;k yo&#x11F;unluk fonksiyonu, veya PDF, kullan&#x131;larak a&#xE7;&#x131;klan&#x131;r. Bireysel sonu&#xE7;lar&#x131;n belirli olas&#x131;l&#x131;klara sahip oldu&#x11F;u ayr&#x131;k da&#x11F;&#x131;l&#x131;mlar&#x131;n aksine, s&#xFC;rekli da&#x11F;&#x131;l&#x131;mlar olas&#x131;l&#x131;klar&#x131; belirli de&#x11F;er aral&#x131;klar&#x131;na tahsis eder. Bir de&#x11F;i&#x15F;kenin belirli bir aral&#x131;&#x11F;a d&#xFC;&#x15F;me olas&#x131;l&#x131;&#x11F;&#x131;, o aral&#x131;k &#xFC;zerinde PDF e&#x11F;risinin alt&#x131;nda kalan alanla temsil edilir. Bu olas&#x131;l&#x131;k integrasyon kullan&#x131;larak hesaplan&#x131;r:\begin{equation*}P(a \leq X \leq b) = \int_{a}^{b} f(x)\, dx\end{equation*}&#xD6;rne&#x11F;in, 150 ile 170 cm aras&#x131;ndaki e&#x11F;ri alt&#x131;ndaki alan 0,75 ise, rastgele se&#xE7;ilmi&#x15F; bir bireyin boyunun bu aral&#x131;kta olma olas&#x131;l&#x131;&#x11F;&#x131; %75&#x2019;tir. Ancak de&#x11F;i&#x15F;kenin herhangi bir tek de&#x11F;eri almas&#x131; olas&#x131;l&#x131;&#x11F;&#x131; her zaman s&#x131;f&#x131;rd&#x131;r; &#xE7;&#xFC;nk&#xFC; tek bir noktan&#x131;n alt&#x131;nda kalan alan s&#x131;f&#x131;rd&#x131;r.Ge&#xE7;erli olabilmesi i&#xE7;in bir olas&#x131;l&#x131;k yo&#x11F;unluk fonksiyonunun t&#xFC;m de&#x11F;erler i&#xE7;in negatif olmamas&#x131; ve de&#x11F;i&#x15F;kenin t&#xFC;m tan&#x131;m alan&#x131; boyunca integrali 1&#x2019;e e&#x15F;it olmas&#x131; gerekir. Bu &#xF6;zellikler, fonksiyonun bir olas&#x131;l&#x131;k da&#x11F;&#x131;l&#x131;m&#x131;n&#x131; do&#x11F;ru bi&#xE7;imde temsil etmesini ve t&#xFC;m m&#xFC;mk&#xFC;n de&#x11F;erler &#xFC;zerindeki toplam olas&#x131;l&#x131;&#x11F;&#x131;n 1&#x2019;e e&#x15F;it olmas&#x131;n&#x131; sa&#x11F;lar.

Olasılık Yoğunluk Fonksiyonlarında İntegrasyonun Uygulamaları

S&#xFC;rekli olas&#x131;l&#x131;k da&#x11F;&#x131;l&#x131;mlar&#x131;, belirli bir aral&#x131;kta herhangi bir ger&#xE7;ek de&#x11F;eri alabilen rastgele ...

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

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Application of Techniques of Integration

İntegral Alma Tekniklerinin Uygulanması

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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Center of Mass in Discrete and Continuous Systems

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Pappus Theorem for Volume and Surface Area

Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

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Consumer Surplus from Demand and Integration

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applications-of-integration-to-find-blood-flow

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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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