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

Ciągłe rozkłady prawdopodobieństwa modelują zmienne losowe, które mogą przyjmować dowolną wartość rzeczywistą w określonym zakresie. Na przykład wzrost dorosłych kobiet może wynosić 163,5, 165,25 centymetra lub dowolną wartość pomiędzy. To sprawia, że wysokość jest ciągłą zmienną losową.Prawdopodobieństwo tej zmiennej opisuje się za pomocą funkcji gęstości prawdopodobieństwa — gładkiej krzywej w zakresie zmiennej. Jednostka tej gęstości jest odwrotną jednostką zmiennej.Całkowite prawdopodobieństwo, że zmienna mieści się w określonym przedziale, ustala się poprzez obliczenie pola pod krzywą dla tego zakresu. Na przykład prawdopodobieństwo, że wzrost kobiety wynosi od 150 do 170 centymetrów, określa się przez całkowanie funkcji gęstości w tym zakresie.Jeśli wynik to 0,75, oznacza to, że 75% kobiet w populacji ma wzrost w tym przedziale.Ale dla dokładnej wysokości prawdopodobieństwo wynosi zero, ponieważ całkowanie nad jednym punktem daje zerową powierzchnię.Prawidłowa funkcja gęstości prawdopodobieństwa jest zawsze nieujemna i całkuje się do 1 w całym zakresie możliwych wartości dla zmiennej losowej.

zastosowania całkowania do funkcji gęstości prawdopodobieństwa

Applications of Integration to Probability Density Functions

Ci&#x105;g&#x142;e rozk&#x142;ady prawdopodobie&#x144;stwa s&#x142;u&#x17C;&#x105; do modelowania zmiennych losowych, kt&#xF3;re mog&#x105; przyjmowa&#x107; dowoln&#x105; warto&#x15B;&#x107; rzeczywist&#x105; w okre&#x15B;lonym przedziale. Zmienne te nie przyjmuj&#x105; warto&#x15B;ci izolowanych ani przeliczalnych, lecz istniej&#x105; na kontinuum warto&#x15B;ci. Na przyk&#x142;ad wzrost danej osoby mo&#x17C;na mierzy&#x107; z rosn&#x105;c&#x105; precyzj&#x105; &#x2013; na przyk&#x142;ad 163,5 lub 165,25 cm &#x2013; co dowodzi, &#x17C;e wzrost jest ci&#x105;g&#x142;&#x105; zmienn&#x105; losow&#x105;.Zachowanie takich zmiennych opisuje si&#x119; za pomoc&#x105; funkcji g&#x119;sto&#x15B;ci prawdopodobie&#x144;stwa, czyli PDF, kt&#xF3;ra okre&#x15B;la spos&#xF3;b rozk&#x142;adu prawdopodobie&#x144;stwa w zakresie mo&#x17C;liwych warto&#x15B;ci. W przeciwie&#x144;stwie do rozk&#x142;ad&#xF3;w dyskretnych, w kt&#xF3;rych poszczeg&#xF3;lne wyniki maj&#x105; okre&#x15B;lone prawdopodobie&#x144;stwa, rozk&#x142;ady ci&#x105;g&#x142;e przypisuj&#x105; prawdopodobie&#x144;stwa przedzia&#x142;om warto&#x15B;ci. Ca&#x142;kowite prawdopodobie&#x144;stwo, &#x17C;e zmienna mie&#x15B;ci si&#x119; w okre&#x15B;lonym przedziale, jest reprezentowane przez pole pod krzyw&#x105; funkcji g&#x119;sto&#x15B;ci w tym przedziale. Oblicza si&#x119; to za pomoc&#x105; ca&#x142;kowania:\begin{equation*}P(a \leq X \leq b) = \int_{a}^{b} f(x)\, dx\end{equation*}Na przyk&#x142;ad je&#x15B;li pole pod krzyw&#x105; mi&#x119;dzy 150 a 170 cm wynosi 0,75, to prawdopodobie&#x144;stwo, &#x17C;e losowo wybrana osoba ma wzrost mieszcz&#x105;cy si&#x119; w tym przedziale, wynosi 75 %. Jednak prawdopodobie&#x144;stwo, &#x17C;e zmienna przyjmie konkretn&#x105;, dok&#x142;adnie okre&#x15B;lon&#x105; warto&#x15B;&#x107;, jest zawsze r&#xF3;wne zeru, poniewa&#x17C; pole odpowiadaj&#x105;ce pojedynczemu punktowi jest r&#xF3;wne zeru.Aby funkcja g&#x119;sto&#x15B;ci prawdopodobie&#x144;stwa by&#x142;a prawid&#x142;owa, musi by&#x107; nieujemna dla wszystkich warto&#x15B;ci oraz musi ca&#x142;kowa&#x107; si&#x119; do 1 w ca&#x142;ej dziedzinie zmiennej. W&#x142;a&#x15B;ciwo&#x15B;ci te zapewniaj&#x105;, &#x17C;e funkcja poprawnie reprezentuje rozk&#x142;ad prawdopodobie&#x144;stwa, a ca&#x142;kowite prawdopodobie&#x144;stwo wszystkich mo&#x17C;liwych wynik&#xF3;w jest r&#xF3;wne 1.

Zastosowania całkowania do funkcji gęstości prawdopodobieństwa

Ci&#x105;g&#x142;e rozk&#x142;ady prawdopodobie&#x144;stwa s&#x142;u&#x17C;&#x105; do modelowania zmiennych losowych, kt&#xF3;re mog&#x105; ...

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

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Application of Techniques of Integration

Zastosowanie technik całkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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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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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