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Suppose that you are told to toss a die until you have
Chapter 5, Problem 88E(choose chapter or problem)
Problem 88E
Suppose that you are told to toss a die until you have observed each of the six faces. What is the expected number of tosses required to complete your assignment? [Hint: If Y is the number of trials to complete the assignment, Y = Y1 + Y2 + Y3 + Y4 + Y5 + Y6, where Y1 is the trial on which the first face is tossed, Y1 = 1, Y2 is the number of additional tosses required to get a face different than the first, Y3 is the number of additional tosses required to get a face different than the first two distinct faces, . . . , Y6 is the number of additional tosses to get the last remaining face after all other faces have been observed. Notice further that for i = 1, 2, . . . , 6, Yi has a geometric distribution with success probability (7 − i )/6.]
Questions & Answers
QUESTION:
Problem 88E
Suppose that you are told to toss a die until you have observed each of the six faces. What is the expected number of tosses required to complete your assignment? [Hint: If Y is the number of trials to complete the assignment, Y = Y1 + Y2 + Y3 + Y4 + Y5 + Y6, where Y1 is the trial on which the first face is tossed, Y1 = 1, Y2 is the number of additional tosses required to get a face different than the first, Y3 is the number of additional tosses required to get a face different than the first two distinct faces, . . . , Y6 is the number of additional tosses to get the last remaining face after all other faces have been observed. Notice further that for i = 1, 2, . . . , 6, Yi has a geometric distribution with success probability (7 − i )/6.]
ANSWER:
Solution :
Step 1 of 1:
Given we are told to toss a die until we have observed each of the 6 faces.
Let Y denotes the number of trials to complete the assignment.
So .
Then has a geometric distribution with success probability
, i=1,2,...,6.