Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / M / 3 / c / Solution

The independent variables $W_i$ have mean two and variance four by part (b). Their standardization isThe central limit theorem therefore proves the claimed convergence to the standard normal law, whose continuous cumulative distribution function is $(2\pi {)}^{-1/2}{\int }_{-\infty }^x{e}^{-u^2/2}du$ at every real $x$.