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

توزيعات الاحتمالات المستمرة تمثل المتغيرات العشوائية التي يمكن أن تأخذ أي قيمة حقيقية ضمن نطاق. على سبيل المثال، قد يكون طول الإناث البالغات 163.5، أو 165.25 سنتيمتر، أو أي قيمة بينهما. وهذا يجعل الارتفاع متغيرا عشوائيا مستمرا.يتم وصف احتمالية هذا المتغير باستخدام دالة كثافة الاحتمالية—وهي منحنى ناعم على مدى المتغير. وحدة هذه الكثافة هي عكس وحدة المتغير.يتم إيجاد الاحتمال الكلي لوقوع المتغير ضمن فترة زمنية محددة بحساب المساحة تحت المنحنى لذلك النطاق. على سبيل المثال، يتم إيجاد احتمال أن يكون طول المرأة بين 150 و170 سنتيمترا عن طريق دمج دالة الكثافة على هذا النطاق.إذا كانت النتيجة 0.75، فهذا يعني أن 75٪ من النساء في السكان لديهن أطوال خلال تلك الفترة.لكن بالنسبة للارتفاع الدقيق، يكون الاحتمال صفرا لأن التكامل فوق نقطة واحدة يعطي مساحة صفرية.دالة الكثافة الاحتمالية الصالحة دائما غير سالبة وتتكامل مع 1 عبر كامل نطاق القيم الممكنة للمتغير العشوائي.

تطبيقات التكامل على دوال كثافة الاحتمال

Applications of Integration to Probability Density Functions

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

Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Application of Techniques of Integration

تطبيقات تقنيات التكامل

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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

arc length of a curve

Measuring Curved Paths with Calculus

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applications-of-integration-to-find-hydrostatic-pressure

Hydrostatic Force on Dam Walls with Integration

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applications-of-integration-to-find-centers-of-mass

Center of Mass in Discrete and Continuous Systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass. For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position,...

theorem of pappus

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 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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