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

连续概率分布用于对可在一定范围内取任意实数值的随机变量进行建模。例如，成年女性的身高可能是 163.5、165.25 厘米，或两者之间的任意数值。这使得身高成为一个连续型随机变量。该变量的概率通过概率密度函数来描述——即在变量取值范围内的一条平滑曲线。该密度的单位是变量单位的倒数。通过计算曲线下特定范围内的面积，可得到变量落在某一特定区间内的总概率。例如，女性身高在150至170厘米之间的概率，可通过在该范围内对密度函数进行积分来求得。如果结果为 0.75，则表示该人群中 75% 的女性身高落在该区间内。但对于一个精确的高度，其概率为零，因为在单个点上积分得到的面积为零。有效的概率密度函数始终为非负值，并且在随机变量所有可能取值的整个范围内积分等于 1。

积分在概率密度函数中的应用

Applications of Integration to Probability Density Functions

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

Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Application of Techniques of Integration

积分技术的应用

KEY TERMS AND CONCEPTS

关键术语与概念

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

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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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