Question

i. Consider a weighted 6-sided die that is twice as likely to produce any even outcome as any odd outcome. What is the expected value of 1 roll of this die? What is the expected value of the sum of 9 rolls of this die?

ii. Let X denote the value of the sum of 10 rolls of an unweighted 6-sided die. What is Pr(X = 0 mod 6)? (Hint: it is sufficient to consider just the last roll) *side note: by X = 0 mod 6, the question is asking when the sum (x) is equal to 0 when you mod 6 (so for a hypethetical example: for 3 rolls if you roll a 3, 5, and 4 that adds up to 12 and 12 mod 6 == 0)

iii. What is the expected number of distinct faces that will be observed in m rolls of an n-sided unweighted die? (Hint: define indicator variable Xi which is 1 if the i’th face is never observed, and use linearity of expectation). You may leave your answer as an expression in terms of m and n.

Answer 1

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