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

התפלגויות הסתברות רציפות מדמות משתנים אקראיים שיכולים לקבל כל ערך ממשי בתוך טווח. לדוגמה, גובה הנקבות הבוגרות יכול להיות 163.5, 165.25 סנטימטרים, או כל ערך ביניהם. זה הופך את הגובה למשתנה אקראי רציף.ההסתברות של משתנה זה מתוארת באמצעות פונקציית צפיפות הסתברות—עקומה חלקה על פני טווח המשתנה. יחידת הצפיפות הזו היא ההופכי של יחידת המשתנה.ההסתברות הכוללת שהמשתנה ייפול בתוך קטע מסוים נמצאת על ידי חישוב השטח מתחת לעקומה עבור אותו טווח. לדוגמה, ההסתברות שגובהה של אישה הוא בין 150 ל-170 סנטימטרים נמצאת על ידי אינטגרציה של פונקציית הצפיפות בטווח הזה.אם התוצאה היא 0.75, זה אומר ש-75% מהנשים באוכלוסייה מחזיקות בגבהים בתקופה זו.אבל עבור גובה מדויק, ההסתברות היא אפס כי אינטגרציה מעל נקודה אחת נותנת שטח אפס.פונקציית צפיפות הסתברותית תקפה היא תמיד לא שלילית ומשתלבת ל-1 בכל טווח הערכים האפשריים עבור המשתנה המקרי.

יישומי אינטגרלציה לפונקציות צפיפות הסתברות

Applications of Integration to Probability Density Functions

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

Education

JoVE Core

Calculus

חשבון דיפרנציאלי ואינטגרלי

Unit

יחידה

Application of Techniques of Integration

יישום של טכניקות אינטגרציה

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

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

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Center of Mass in Discrete and Continuous Systems

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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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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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