Consider a discrete rv X with the PMF

pX (-1) = (1 - 10-10)/2,

pX (1) = (1 - 10-10)/2,

pX (1012) = 10-10.

(a) Find the mean and variance of X. Assuming that {Xm; m ≥ 1} is an IID sequence with the distribution of X and that Sn = X1 + ··· + Xn for each n, find the mean and variance of Sn. (No explanations needed.)

(b) Let n = 106 and describe the event {Sn ≤ 106} in words. Find an exact expression for PrfSn ≤ 106l = FSn (106).

(c) Find a way to use the union bound to get a simple upper bound and approximation of 1 - FSn (106).

(d) Sketch the CDF of Sn for n = 106. You can choose the horizontal axis for your sketch to go from -1 to +1 or from -3 × 103 to 3 × 103 or from -106 to 106 or from 0 to 1012, whichever you think will best describe this CDF.

(e) Now let n = 1010. Give an exact expression for PrfSn ≤ 1010l and show that this can be approximated by e-1. Sketch the CDF of Sn for n = 1010, using a horizontal axis going from slightly below 0 to slightly more than 2 × 1012. Hint: First view Sn as conditioned on an appropriate rv.

(f) Can you make a qualitative statement about how the CDF of a rv X affects the required size of n before the WLLN and the CLT provide much of an indication about Sn.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.