Question 2.51: Let Xi, i = 1, 2, ... , 10 be independent random variables, ......
Let X_{i}, i = 1, 2, … , 10 be independent random variables, each being uniformly distributed over (0, 1). Estimate P\{\textstyle\sum_{1}^{10}X_{i}\gt 7\}.
Since E[X_{i}]={\frac{1}{2}},\mathrm{Var}(X_{i})={\frac{1}{12}} we have by the central limit theorem that
P\left\{\sum_{1}^{10}X_{i}\gt 7\right\}=P\left\{\frac{\sum_{1}^{10}X_{i}-5}{\sqrt{10\left(\frac{1}{12}\right)}}\gt \frac{7-5}{\sqrt{10\left(\frac{1}{12}\right)}}\right\}\approx1-\Phi(2.2)
= 0.0139
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