Probability of sum of uniform variables

Question

I need to estimate the probability of a sum of uniform variables. More exactly $P(S_{100} \geq 70)$ where $S_{100}=\sum_{i=1}^{100} X_i$ and $X_1, X_2, \cdots, X_{100}$, are iid with $\mathcal{U}$(0,1) distribution. I find solution for other distribution but can't apply it for variables with uniform distribution. I want to use central limit theorem and struggle to write the probability with an expression of sum and pdf (if possible). Please help.

Answer 1

HINT - CLT:

For uniform distribution $U_X(0,1): \ \ E(X) = \frac{1}{2},\ \ D(X) = \frac{1}{12}$

For sum of 100 variables $ U_X(0,1): \ \ E(S_{100})=100\cdot\frac{1}{2} = 50, \ \ D(S_{100})=100\cdot \frac{1}{12}=\frac{25}{3}$

Then $P(S_{100} \geq 70)=1-P(S_{100} < 70)\sim 1-F(70), \ \ F(x) = \ $ distribution function of normal distribution $N(\mu,\sigma^2) =N(50, 25/3).$

Answer 2

Use the Irwin-Hall distribution: https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution

Answer 3

Actually it is Irwin-Hall disrtribution with pdf \begin{eqnarray*} f_{\sum_{i=1}^{100} X_i}(s)&=&\frac{1}{99!}\sum_{k=0}^{\lfloor s\rfloor}(-1)^k\binom{100}{k}(s-k)^{99}, \; \; \; 0\leq s\leq 100\\ &=& 0, \; \; \; \mathrm{elsewhere}, \end{eqnarray*} but it is difficult, you can use central limit theorem afterall (fyi: irwin-hall distribution actually is used to approximate normal distribution tables)