Uniform distribution ($U(a,b)$)

A continuous uniform distribution has constant density on its interval of support.

For independent $U,V$ uniform on $[0,1]$, real roots occur on $U\le V^2$, a region of probability $1/3$. On that region both roots are nonpositive. Both lie in $[-1,0]$ exactly when $\max (0,2V-1)\le U\le V^2$. Its area is $1/12$, so the conditional probability of both absolute values being at most one is $1/4$. Joint probability density integration turns algebraic discriminant and root constraints into simple planar areas.

A uniform random variable has a uniform distribution. Independent such variables can supply independent inverse transform sampling; reusing a single variable for two transforms generally creates dependence.

A Steinhaus random variable is uniformly distributed on the complex unit circle. It can be generated as $ϵ\cos \theta +i\delta \sin \theta$ using independent signs $ϵ,\delta$ and an independent uniform angle $\theta \in [0,\pi /2]$. This conditional sign representation transfers Rademacher sum inequalities to random complex phases.

For independent Steinhaus random variables, condition on their acute angles and use the sharp Rademacher second-moment inequality on the $2d$ sign coefficients $a_i\cos {\theta }_i,i{a}_i\sin {\theta }_i$. Their squared moduli sum to ${\sum }_i|a_i{|}^2$ independently of the angles, so the conditional first moment is at least the same fixed square-root bound. Averaging proves the assertion.

The square of a uniform distribution variable on $(0,1)$ has cumulative distribution function $\sqrt{x}$ and an integrable inverse-square-root probability density function. Reflection replaces $x$ by $1-x$ in that density.

For independent $U_1,U_2\sim \mathrm{Uniform}(0,1)$, the sum has triangular densityIts upper tail for $1\le c\le 2$ is $(2-c{)}^2/2$.

Let $X\ge 0$ have density $g$, let $U$ be independent and uniform on $[0,1]$, and setThen

Write $h(y)={\int }_y^{\infty }g(t)/tdt$. The two pieces in a uniform split are independent exactly whenSetting $z=0$ shows that $h(y)=c{e}^{-cy}$ for some $c>0$. Thus the pieces are independent exactly when they are independent exponential variables of the same rate; equivalently, the total has gamma density $c^{2}x{e}^{-cx}$.