Continuous distributions and density functions
| English | 中文 | Pinyin |
|---|---|---|
| probability density/ˌprɒbəˈbɪlɪti ˈdensɪti/ | 概率密度 | gài lǜ mì dù |
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?
A nonnegative function whose integral over an interval gives its probability.
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
A density f must be nonnegative and integrate to 1 over its support
Compare the model with the worked case and explain one change.
Find k when f(x)=kx on [0,2] is a density.
Total density area k[x²/2] from 0 to 2=2k=1, so 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).
Probability=integral x/2 from 0 to 1=[x²/4]=1/4.
A continuous variable has P(X=x) equal to the density height 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.
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).
E(X)=integral x²/2 from 0 to 2=[x³/6]=4/3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
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.