Please answer question 1, 2, 6, 7 step by step and provide some explanation

(1) [35 marks] Suppose n balls are thrown randomly into m boxes. Each ball lands in each box with uniform probability. Dene X,- be the r.v. equal to the number of balls that land in box i. o What is the distribution of X,? Compute IE[X,~] and Var[X,]. [15 marks] - Are the X, r.v's (i) mutually independent (ii) pairwise independent? Justify your reasoning. [5 marks] I For m = 500, n = 1000, using the Chernoff bound, prove that, Pr[X,~ (x + a)2. Using this reasoning, . Apply the Markov bound to the r.v Y, and prove the following statement: Pr[R - E[R] > x] a" + Var [R] ( a + 2 ) 2 [15 marks] . Prove the one-sided Chebyshev's Theorem by finding the best value of a (optimize w.r.t a to obtain the tightest bound). [10 marks]