4.1 Continuous random variables

If a rod has linear density ρ(x), its mass on [a,b] is ∫ₐᵇρ(x)dx. A piecewise density uses the appropriate expression on each subinterval, and masses or centres require separate integrals.

Check continuity or a jump at the boundary, split the integral exactly there, then form x̄=(1/M)∫xρ(x)dx. Units of density must match the coordinate units.

If ρ=2 on 0≤x<1 and ρ=4 on 1≤x≤3, total mass is 2(1)+4(2)=10; the centre uses the corresponding weighted first moments.

A density value is not a mass, and using one formula across a discontinuity silently changes the physical model.

For a discrete random variable X, E(X)=ΣxP(X=x). It is the long-run mean of repeated trials and may be a value that no single outcome can take.

Check that probabilities sum to one before calculating. For a function g(X), use E[g(X)]=Σg(x)P(X=x), not generally g(E(X)).

If X is 0 or 2 with probabilities 0.7 and 0.3, E(X)=0.6; no individual trial has outcome 0.6. But over many trials the average approaches 0.6.

Expected value is not the most likely value and is not a promise about the next observation.

For a continuous variable, the probability density function f(x) satisfies P(a≤X≤b)=∫ₐᵇf(x)dx, while the cumulative distribution function F(x)=P(X≤x)=∫_{−∞}ˣf(t)dt.

A density value is not itself a probability; probabilities are areas. The CDF is non-decreasing, approaches 0 and 1 at the ends, and where differentiable F′(x)=f(x).

If f(x)=2x on 0≤x≤1, then P(X≤0.5)=∫₀^.5 2x dx=0.25 and F(0.5)=0.25.

For a continuous variable P(X=x)=0, even though f(x) may be positive; use an interval or tail probability.

Once F is known, P(a<X≤b)=F(b)−F(a), P(X>b)=1−F(b), and quantiles are found by solving F(x)=p. Endpoints do not matter for continuous distributions.

State the support before evaluating a CDF. If a piecewise formula is given, use the correct branch and check continuity at the join.

If F(x)=x² on 0≤x≤1, then P(0.2<X≤0.8)=0.64−0.04=0.60; the density is not needed for this subtraction.

F(b)−F(a) is not F(b−a), and a CDF value is cumulative probability, not a density height.