Question 8: You roll a fair die repeatedly, and independently, until you have se

Question

Question 8: You roll a fair die repeatedly, and independently, until you have seen all of the numbers 1,2,3,4,5, 6 at least once. Define the random variable Xthe number of times you roll the die. For example, if you roll the sequence 5,5,3,5,1,3, 4,2,5,2, 1, 3,6, then X 13. Determine the expected value E(X) of the random variable X. Hint: Use the Linearity of Expectation. If you have seen exactly i different elements from the set 1,2,3, 4,5,6], how many times do you expect to roll the die until you see a nevw element from this set?

Explanation / Answer

Here have to find the expected roll to see that atleast all numbers come on the roll.

So, in first roll there will be a number, a unique number. So, the probability that a unique number will come on first roll =1

In second roll, the probability that a unique number will come = 5/6

as we know after first roll, the probability that the unique number will not come = 1/6

so the expectation of unique number = 1/ (1 - 1/6) = 6/5

similarly, when two unique number has come then the probability that the unique number will not come = 2/6

so the expectation of a uique number on further roll = 1/ (1 - 2/6) = 6/4

so similarly,

E(X) = 1 + 1/(5/6) + 1/(4/6) + 1/(3/6) + 1/(2/6) + 1/(1/6) = 1 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.7

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