Question 7: Let m 1 and n 1 be integers. You are given m cider bottles C1,C2,.,Cm and n beer bottles B1, B2,, B Consider a uniformly random permutation of these m+n bottles. The positions in this permutation are numbered 1,2,...,m+n. Define the random variable X = the position of the leftmost cider bottle Determine the possible values for X For any value k that X can take, prove that m (-1) m+n Hint: Use the Product Rule to determine the number of permutations for which X = k Rewrite your answer using basic properties of binomial coefficients For each i-1,2,... , n, define the indicator random variable 1 if Bi is to the left of all cider bottles, i0 otherwise Prove that E (X) = m + 1 Express X in terms of X1, X2,..., X, Use the expression from the previous part to determine E(x) Prove that mn+1 k-1 Question 7: Let m 1 and n 1 be integers. You are given m cider bottles C1,C2,.,Cm and n beer bottles B1, B2,, B Consider a uniformly random permutation of these m+n bottles. The positions in this permutation are numbered 1,2,...,m+n. Define the random variable X = the position of the leftmost cider bottle Determine the possible values for X For any value k that X can take, prove that m (-1) m+n Hint: Use the Product Rule to determine the number of permutations for which X = k Rewrite your answer using basic properties of binomial coefficients For each i-1,2,... , n, define the indicator random variable 1 if Bi is to the left of all cider bottles, i0 otherwise Prove that E (X) = m + 1 Express X in terms of X1, X2,..., X, Use the expression from the previous part to determine E(x) Prove that mn+1 k-1