8. Let Xi, i= 1,2,... be a sequence of independent and identically distributed random variables with P(X; = 1) = P(X; = -1) = 1/2. A random walk is created by starting at position 0 and then consecutively moving forward or backward according to X1, X2, . ... Let Y; denote the position of the walker at time j, which is Yj = _Xi. i=1 Answer the following questions about the walker's position at time 100, Y100. For convenience, we will write Y100 = Y. (a) Find the characteristic function for Xi (b) Find the Chebyshev bound (inequality) for P(|Y| 2 30) (c) Find the characteristic function for Y. (d) Find the Chernoff bound for P(Y 2 30) (e) Find the CLT approximation for P(Y > 30) (f) Find the true value for P(Y > 30). We can write X; as Xi = 2B; - 1. Use this idea to write P(Y > 30) as some probability on B. I recommend you use the sf () or cdf () method of a binomial distribution created using scipy . stats . binom to evaluate the probability