Module 24: Improper integrals and probability density

A. Prerequisites and observable goals#

B. The central idea: accumulation can extend beyond finite, bounded data#

The Riemann integral on a compact interval was built for bounded functions. Infinite intervals or unbounded integrands require limits of ordinary integrals. A small tail may accumulate to a finite total or an infinite one. The integrand tending to zero is not enough: $1/x$ tends to zero but its tail integral diverges.

C. Definitions and comparison tools#

For continuous $f$ on every finite subinterval of $[a,\infty)$,

if the limit is finite. At an unbounded endpoint $a$, use $\lim_{c\to a^+}\int_c^b f$. At an interior singularity, split into two independent improper integrals and require both to converge. On $(-\infty,\infty)$, split at a finite point and require each tail separately. A finite symmetric limit can be a Cauchy principal value without being an ordinary improper integral.

For nonnegative functions, if $0\le f\le g$ eventually and $\int g$ converges, then $\int f$ converges; if $\int f$ diverges, then $\int g$ diverges. Finite initial pieces do not change tail convergence. For $p$-integrals,

Different endpoint regimes produce opposite inequalities. [O2Improper]

A probability density $p(x)$ must be nonnegative and have total integral one over its support. The probability of an interval is the integral of the density there, not a point height. When $\int|x|p(x)\dd x<\infty$, the mean is $E[X]=\int xp(x)\dd x$. A normalized density need not have a finite mean. If $x$ has length units, a density has reciprocal-length units.

D. Why the exponent threshold is one#

For $p\ne1$,

If $p>1$, the exponent $1-p$ is negative and $b^{1-p}\to0$, leaving $1/(p-1)$. If $p<1$, it grows without bound. At $p=1$, the primitive is $\ln b$, which also diverges. At zero the same power expression is examined as its lower input approaches zero, reversing the exponent condition. The convergence test comes from a limit, not simply the sign of $p$.

E. A procedure for improper problems#

Mark every infinite bound and singular point before integrating. Split wherever necessary. Replace one problematic endpoint at a time with a finite variable and calculate an ordinary integral. Take each limit independently. For a convergence-only question, comparison may be quicker than finding a primitive. For a density, verify nonnegativity, then normalization, then any requested probability or moment; do not assume a moment exists just because normalization succeeds.

F. Worked examples#

An exponential tail. For $\lambda>0$, $\int_0^b\lambda e^{-\lambda x}\dd x=1-e^{-\lambda b}\to1$, so this function is a density on $x\ge0$. Integration by parts gives

The vanishing product $be^{-\lambda b}$ follows from exponential dominance, not from assigning a value to $\infty\cdot0$.

A misleading symmetry. The symmetric truncated integral of $1/x$ from $-b$ to $-\varepsilon$ and from $\varepsilon$ to $b$ is zero. Nevertheless its two near-zero pieces diverge independently. Symmetry can coordinate cancellations but cannot change the definition of an ordinary improper integral.

G. Independent practice#

F foundation C standard course work T transfer S optional stretch L runnable lab or project. Attempt the complete set before opening its solutions.

H. Complete solutions: Module 24#

Compare the first step where your work differs. A corrected answer is not yet an independent success.

Open every solution

I. Retention and readiness#

Revisit M15.05, M22.08, and M19.07. Explain why convergence, numerical value, and existence of a moment are separate questions. Advance when you can list every independent limiting endpoint before performing any antiderivative calculation.