6. (8 pts) A twelve-sided die, with the equally-likely outcomes 1,2,3,. . . 12,

Question

6. (8 pts) A twelve-sided die, with the equally-likely outcomes 1,2,3,. . . 12, is tossed until a is obtained. Let x, be the number of tosses required. This process is repeated a second time with x, the number of tosses required to get a second ‘l,' and so on. Altogether, the die is tossed until a" 1, appears n different times with the associated required number of tosses being x,, , Let izl Then, for or n not small: a. F should be close to what? (Give a mumber.) b. s should be close to what? (Give a mumber.)

Explanation / Answer

(a) Here as we know that Pr(getting 1 on a 12 sided die) = 1/12

Here to get the first 1 on the die unitl 1 is obtained, the distribution becomes the geometric distribution which have expected value E(x) = 1/1(1/12) = 12

so as we repeat the process numerous times, we will get sample mean ,

x = 1/(1/12) = 12

Here same is for sample standared deviation as it is the point estimate of population standard deviation

s = sqrt [(1-p)/p2] = sqrt [(1 - 1/12)/(1/12)2] = sqrt [(11/12)/ (1/12)2] = 11.49

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