Uniform sum distribution.

**lohengrin** · April 1st 2012, 05:34 AM

Hi,
I have a problem with calculating sum of n uniform variables on the interval [0,1].

I found solution here:
Uniform Sum Distribution -- from Wolfram MathWorld
I was thinking about charakteristic function but I do not understand one line:

Assume we calculated characteristic function of that sum and we obtained $(\frac{i(1-e^{i t})}{t})^n$ .
Then we want to calculate probability density ( using $P(x)= \frac{1}{2 \Pi} \int_{- \infty}^ {\infty} e^{- i t x } \phi(t) dt$

But I do not know how calculate: $P(x)= \frac{1}{2 \Pi} \int_{- \infty}^ {\infty} e^{- i t x } (\frac{i(1-e^{i t})}{t})^n dt$

Please give me a hint

**Alibiki** · April 2nd 2012, 02:14 PM

If the integer n is greater than 3, then you may safely invoke the Central limit theorem and claim that the exact probability distribution of the sum resembles a certain normal probability distribution; if you want to investigate the rate of convergence to the normal distribution, I suggest taking a look at the Berry-Essén theorem.

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