Standard deviation

[squenshl]

Consider the Uniform distribution (continuous) U(-1,1) which has p.d.f. f(x) = 1/2 for -1 < x < 1 and 0 otherwise.

I have calculated the Fourier transform using the characteristic function and got fhat(epsilon)= sin(epsilon)/epsilon

How do I calculate the standard deviation of this distribution using both the Fourier transform and the definition. I know that the standard deviation is 1/sqrt(3) but how do I get this using whats above.

[mathman]

The simplest method is to calculate the integral of x2 with respect to the density function you are given (=1/3). Since the mean = 0, the variance = 1/3 and the standard deviation = 1/√3. You could expand fhat into a power series and the coefficient of the x2 term gives the second moment (there may be a factor here).

[bpet]

By definition fhat(t)=E[exp(itX)] so if you expand exp(itX) in a Taylor series you'll get an expression for the moments in terms of the Taylor series of fhat.