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I'm having some troubles with an exercise of statistics. I must find the mean and the variance of the gaussian distribution, and for this I intend to use the characteristic function and its natural logaritmic.

My professor sad that the characteristic function is $$p(k) =\frac{1}{\sqrt{2\pi\sigma^2}}\int_{\Bbb R} e^{{\frac{-({x-\mu})^2}{2\sigma^2}}-{ikx}}dx = ... = e^{-ik\mu - \frac {k^2\sigma^2}{2}}$$

but I cannot express this integral as this exponential.

My atempts:

I tried to express the argument of the exponent as a perfect square to use the result of the integral of $e^{-x^2}$ but the answer was enormous.I also tried to solve the integral using integration by parts, but I failed to find the correct form for the answer.

Can someone give me a hint?

  • $\begingroup$ You cannot make substitutions like $y=x+c$ with $c$ complex without bringing in Complex Analysis. Calculus does not allow that. If you do not know Complex Analysis forget about completing squares to get the answer. You can use a DE approach but that also requires theorems on interchange of integrals and derivatives. A standard approach is to use contour integration from Complex Analysis. A power series approach is also possible. Just downvoting every answer is not going to help you. $\endgroup$ – Kavi Rama Murthy Jun 23 '19 at 23:48

Characteristic function involve complex valued functions and some knowledge of Complex Analysis would; be useful. Of the many ways in which this characteristic function can be found here is the one I like the most: if $t$ is real then $\int e^{-x^{2}/2} e^{tx}dx=\sqrt {2\pi}e^{t^{2}/2}$; you can show by completing the square in $x^{2}/2-tx$. Once you do this Complex Analysis can be used to show that you can simply replace the real number $t$ by $ik$ where $k$ is real. [If two entire functions coincide on the real line they coincide on the complex plane. Of course, we have to prove that the left side is an entire function. This can be proved using DCT]. This gives you the characteristic function of satndard normal distribution and you can use a simple change of variable when the mean is $\mu$ and variance is $\sigma^{2}$.


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