The Δ²/12 noise variance

ProofBeginner⏱ 15 minQuantizationvarianceuniformnoise model

Problem

Prove that a uniform random variable on $[-\Delta/2,\Delta/2]$ has variance $\Delta^2/12$ .
Prove that if $X$ is uniform on a single quantization cell then the error is exactly uniform.
Give an explicit distribution for $X$ for which the error has variance strictly greater than $\Delta^2/12$ , and one for which it is zero.

Write a complete argument. Every inequality you use should be one you could state precisely.

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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