A stock begins at price 2000 and each day moves an amount th

A stock begins at price 2000 and each day moves an amount that is uniformly distributed over [-2,4].The movement from day to day are independent. Let X be the price of the stock after 300 days. Use markov\'s inequality to get a bound on P(x ge 2360) use the control limit theorem to give an approximation to P(x ge2360).

Solution

a) The range after 200 days is [2000+300*-2,2000+300*4] or [1400, 3200], so x is non-negative

For a uniform distribution from -2 to 4, the mean is (-2+4)/2 = 1

Then, the sum of 300 rv with mean 1 has mean 300*1 = 300, so x\'s mean is 2000+300 = 2300

Then, Markov\'s inequality is that, for a non-neg rv, P(X>=a) <= E(x)/a, so

P(X >= 2360) <= E(x)/2360 = 2300/2360 = 115/118 = .9746

b) If Y ~ U[-2,4], then var(y) = (b-a)^2/12 = (4- -2)^2/12 = 36/12 = 3

Then, the sum of 300 independent rv, each ~ U[-2, 4], has variance 300 * var(y) = 300 *3 = 900

Then, as E(x) = 2300 and var(x) = 900

P(X >= 2360) = P(z >= (2360-2300)/sqrt(900)) = P(z >= 60/30) = P(Z >= 2) = 1 - phi(2) =

from Excel, 1 - normsdist(2) = .0228