6.3 Continuous random variables

For a continuous X, probabilities are areas over intervals: P(a≤X≤b)=∫ₐᵇf(x)dx, and P(X=x)=0 for every single point.

State the support and use a density or CDF that is non-negative and normalised. Endpoint inclusion does not change a continuous probability.

If X is uniform on [0,4], P(1<X<3)=2/4=0.5.

A density height is not a probability; only an area over a range is.

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.