The sum of uniform draws

Topic: Functions of random variables and limit theorems · more questions on this topic
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Numbers are drawn independently from the uniform distribution on [0,1], and S_n is their sum.

  1. Describe the density of S_2.

  2. Give the normal approximation to the distribution of S_{20}.

  3. Approximate the probability that the average of the 20 numbers exceeds 0.6. Use \Phi(1.55)\approx 0.9394.

  1. The density of S_2 is the triangular function f(s)=s for 0<s<1 and f(s)=2-s for 1\le s<2, and zero elsewhere. It rises linearly from 0 to 1 and falls linearly from 1 to 2.

  2. Each uniform draw has mean 1/2 and variance 1/12, so S_{20} is approximately \operatorname{N}(10, 20/12).

  3. The average exceeds 0.6 when S_{20}>12. With Y\sim\operatorname{N}(10,20/12),

\mathbb{P}(Y>12)=1-\Phi\bigl((12-10)/\sqrt{20/12}\bigr)=1-\Phi(1.549)\approx 1-0.9394=0.0606.