6.3.2—Density functions
- Syllabus
- 9709–2028–2029
- Objective
- 6.3.2
- Level
- A2
A density function must be non-negative and integrate to one
A PDF f(x) satisfies f(x)≥0 and ∫f(x)dx over its support=1. Interval probabilities are integrals of f, and the CDF is their accumulated area.
Find an unknown constant by normalising, then use the correct support for probabilities and moments. A density may exceed 1 when its units are inverse-length.
If f(x)=kx on 0≤x≤2, normalisation gives k=1/2; then P(X>1)=∫₁²x/2 dx=3/8.
A PDF value of 1.5 is not impossible; probabilities are areas and the total area, not the peak height, is constrained.
ConceptA-Level CAIE Mathematics A2