Continuous distributions and density functions · Distribuciones continuas y funciones de densidad
| English | Español |
|---|---|
| probability density/ˌprɒbəˈbɪlɪti ˈdensɪti/ | densidad de probabilidad |
Is a height the same as a probability?
- A waiting time can take any value in an interval. Its probability comes from area, not the graph's height at one instant.
- This lesson studies probability density 概率密度: A nonnegative function whose integral over an interval gives its probability.
Choose the mathematical structure
- A density f must be nonnegative and integrate to 1 over its support. The cumulative distribution F(x) is the integral up to x. For a continuous variable P(X=x)=0 and E(X) is the integral of xf(x).
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines probability density? · ¿Cuál descripción define correctamente la densidad de probabilidad?
A nonnegative function whose integral over an interval gives its probability. · Una función no negativa cuya integral sobre un intervalo da su probabilidad.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Let f(x)=kx on 0≤x≤2 and zero elsewhere. Normalisation gives integral_0^2 kx dx=2k=1, so k=1/2. Then P(X≤1)=integral_0^1 x/2 dx=1/4 and E(X)=integral_0^2 x²/2 dx=4/3.
Continuous distributions and density functions · Distribuciones continuas y funciones de densidad
A density f must be nonnegative and integrate to 1 over its support · Una densidad f debe ser no negativa e integrar a 1 sobre su soporte
Compare the model with the worked case and explain one change. · Compara el modelo con el caso resuelto y explica un cambio.
Find k when f(x)=kx on [0,2] is a density. · Encuentra k cuando f(x)=kx en [0,2] es una densidad.
Total density area k[x²/2] from 0 to 2=2k=1, so k=1/2. · Área total de densidad k[x²/2] desde 0 hasta 2=2k=1, por lo tanto k=1/2.
Test a tempting shortcut
- Density can exceed 1 without being invalid; total area must equal 1. A cumulative distribution cannot decrease. Include the zero-density region outside the support when defining a complete model.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A continuous variable has P(X=x) equal to the density height f(x). This claim is false. Explain which definition or assumption it violates.
For that density, find P(X≤1). · Para esa densidad, encuentra P(X≤1).
Probability=integral x/2 from 0 to 1=[x²/4]=1/4. · Probabilidad=integral x/2 desde 0 hasta 1=[x²/4]=1/4.
A continuous variable has P(X=x) equal to the density height f(x). · Una variable continua tiene P(X=x) igual a cero; la probabilidad se obtiene del área bajo la curva de densidad f(x).
Density can exceed 1 without being invalid; total area must equal 1. A cumulative distribution cannot decrease. Include the zero-density region outside the support when defining a complete model. · La densidad puede exceder 1 sin ser inválida; el área total debe ser igual a 1. Una función de distribución acumulada no puede disminuir. Incluye la región de densidad cero fuera del soporte al definir un modelo completo.
Interpret a new situation
- For a uniform distribution, density is reciprocal interval length. Find medians or percentiles using the cumulative probability, and use a support-aware integral for moments.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For that density, find E(X). · Para esa densidad, encuentra E(X).
E(X)=integral x²/2 from 0 to 2=[x³/6]=4/3. · E(X)=integral x²/2 desde 0 hasta 2=[x³/6]=4/3.
Match each part of a complete solution to its purpose. · Emparejar cada parte de una solución completa con su propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Una suposición justifica el modelo; una comprobación verifica el resultado; la interpretación lo conecta con la pregunta.
Use this in your course
- edexcel IAL further mathematics; official unit S2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A nonnegative function whose integral over an interval gives its probability. Choose the relationship, show the method, check its assumptions and interpret the result.