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

連続確率分布は、任意の実数値を範囲内で取ることができる確率変数をモデル化します。例えば、成虫の雌の身長は163.5センチ、165.25センチメートル、またはその間の任意のサイズかもしれません。これにより高さは連続的な確率変数となります。この変数の確率は確率密度関数、すなわち変数の範囲上の滑らかな曲線を用いて記述されます。この密度の単位は、変数の単位の逆数です。変数が特定の区間内に収まる確率は、その範囲の曲線下面積を計算することで求められます。例えば、女性の身長が150センチから170センチメートルの間である確率は、その範囲の密度関数を積分することで求められます。もし結果が0.75であれば、人口の75%の女性がその区間内で身長を持っていることを意味します。しかし正確な高さの場合、1点で積分すると面積がゼロになるため、確率はゼロになります。有効な確率密度関数は常に負ではなく、確率変数の可能な値全範囲にわたって1に積分されます。

確率密度関数への積分の応用

Applications of Integration to Probability Density Functions

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

Education

JoVE Core

Calculus

微積分学

Unit

単位

Application of Techniques of Integration

積分法の応用

KEY TERMS AND CONCEPTS

主要用語と概念

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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