It is generally not possible to express p_{\overline X_n}(x) in a closed form given an arbitrary PDF p_X^{\phantom X} and a number n . But for the limit n\to\infty it is possible to make an approximation. The very important result is called the central limit theorem. It tells us that as n goes to infinity, p_{\overline X_n}(x) approaches a Gaussian distribution whose mean and variance equal the true mean and variance, \mu_{X}^{\phantom X} and \sigma_{X}^{2} , respectively:
\begin{equation} \lim_{n\to\infty} p_{\overline X_n}(x) = \left(\frac{n}{2\pi\mathrm{var}(X)}\right)^{1/2} e^{-\frac{n(x-\bar x_n)^2}{2\mathrm{var}(X)}} \tag{12} \end{equation}