Section 8.1

Sums, Products and Ratios of Uniforms

Functions of independent uniforms by convolution or area: sums, products and their densities, ratios, floor and ceiling events, roots of random equations.

Toolkit

For independent uniform (or related) variables, compute probabilities and densities of sums, products, ratios, and floor/log transforms geometrically or by conditioning on one variable and integrating over the other.

Key tools:

Area method: for $X,Y\sim\mathrm{Unif}(0,1)$ independent, $\mathbb{P}((X,Y)\in A)=\mathrm{area}(A)$ ; ratio events $X/Y>m$ are wedges through the origin, giving $\mathbb{P}(X/Y>m)=\frac{1}{2m}$ for $m\ge1$ .
Conditioning: $\mathbb{P}(E)=\int_0^1 \mathbb{P}(E\mid X=x)\,dx$ ; if the conditional probability is linear in $x$ , only $\mathbb{E}[X]$ matters.
Simplex volumes: $\mathbb{P}(U_1+\cdots+U_n<1)=\dfrac{1}{n!}$ .
Log transform: $-\ln U\sim\mathrm{Exp}(1)$ ; sums of exponentials are Gamma, so $-\ln(U_1\cdots U_n)\sim\mathrm{Gamma}(n,1)$ , giving density $\dfrac{(-\ln y)^{n-1}}{(n-1)!}$ for the product on $(0,1)$ .
Law of large numbers via logs: $\mathbb{E}[\ln U]=-1$ controls geometric means.
Telescoping/zeta series arise when summing dyadic or ratio-band probabilities.
Rayleigh-type integrals: $\int_0^\infty b\,e^{-b^2/2}(\cdots)\,db$ often integrates in closed form.

Problems