Density of the sum of two independent uniforms

Question

If $X$ and $Y$ are independent and each uniform on $[0,1]$ , what is the probability density function of $Z=X+Y$ ?

Short answer

Triangular: $f_Z(z)=z$ on $[0,1]$ , $2-z$ on $[1,2]$ , and $0$ elsewhere. Convolve the two densities: $f_Z(z)$ is the length of the set of $x\in[0,1]$ with $z-x\in[0,1]$ .

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q3.33, p.84; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Independence
Cumulative Distribution Functions

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