Problem 3 (3 points)

Random Walk theory provides us with the following nice (anf maybe counterintuitive) asymptotic statements:

limN[C2N2Nx]=2arcsin(x)limNP[C2N2Nx]=2arcsin(x)

limN[L2N2Nx]=2arcsin(x)limNP[L2N2Nx]=2arcsin(x)

limN[M2N2Nx]=2arcsin(x)limNP[M2N2Nx]=2arcsin(x)

We say that the random variables C2N/2N,L2N/2N,M2N/2NC2N/2N,L2N/2N,M2N/2N converge in distribution to the Arcsine Distribution.

The interesting property about the Arcsine distribution is that its density (see its formula above) is U-shaped on (0,1)(0,1). In other words, if XX is arcsine-distributed on (0,1)(0,1), the probabilty that XX takes very small values near 0 or very large values near 1 is rather high, but the probability for taking values around, say, 0.5, is low.

For 2N=10002N=1000 sample 10,000 realisations of each of the random variables C2N/2N,L2N/2N,C2N/2N,L2N/2N, and M2N/2NM2N/2N, respectively. Display a normalized histogram for all three simulations, along with the probability density function of the arcsine distribution, to check the above facts numerically!