Question 7: Let m > 1 and n > 1 be integers. You are given m cider bottles C1, C2, , Cm and n beer bottles Bi, B2,... , Bn. 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+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 B, is to the left of all cider bottles, 0 otherwise ?. Prove that E (X) = m + 1 Express X in terms of Xi,X2,.. . ,Xn. Use the expression from the previous part to determine E(X) Prove that m+n+1 n+i (n Question 7: Let m > 1 and n > 1 be integers. You are given m cider bottles C1, C2, , Cm and n beer bottles Bi, B2,... , Bn. 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+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 B, is to the left of all cider bottles, 0 otherwise ?. Prove that E (X) = m + 1 Express X in terms of Xi,X2,.. . ,Xn. Use the expression from the previous part to determine E(X) Prove that m+n+1 n+i (n