Consider the experiment of rolling a fair die independently until the same number/face occurs 2 successive times and let X be the trial on which the repeat occurs, e.g. if the rolls are 2,3,4,5,1,2,4,5,5, then&nbsp;X&nbsp;=&nbsp;9&nbsp;.&nbsp;a. find the probability function&nbsp;f&nbsp;(&nbsp;x&nbsp;)&nbsp;=&nbsp;P&nbsp;(&nbsp;X&nbsp;=&nbsp;x&nbsp;)&nbsp;b. compute EX

Question

Consider the experiment of rolling a fair die independently until the same number/face occurs 2 successive times and let X be the trial on which the repeat occurs, e.g. if the rolls are 2,3,4,5,1,2,4,5,5, then&amp;nbsp;X&amp;nbsp;=&amp;nbsp;9&amp;nbsp;.&amp;nbsp;a. find the probability function&amp;nbsp;f&amp;nbsp;(&amp;nbsp;x&amp;nbsp;)&amp;nbsp;=&amp;nbsp;P&amp;nbsp;(&amp;nbsp;X&amp;nbsp;=&amp;nbsp;x&amp;nbsp;)&amp;nbsp;b. compute EX

ehedem26 · Accepted Answer

Step 1This is basically a geometric distribution. At each step, independent of whatever happened previously, you have a 1/6 chance of rolling the same as you did on the previous try. You keep going until you get a success, and you are counting the number of tries until a success. Let&#039;s call this random variable Y.Step 2Now here comes the only twist: You start not on the first try (on which it is impossible to duplicate a previous roll), but on the second. So X is one more than Y, hence       f    X    (  t  )  =      f    Y    (  t  −  1  ), and   E  (  X  )  =  E  (  Y  +  1  )

Marilyn Cameron · Accepted Answer

Step 1    This is basically a geometric distribution. At each step, independent of whatever happened previously, you have a 1/6 chance of rolling the same as you did on the previous try. You keep going until you get a success, and you are counting the number of tries until a success. Let&#039;s call this random variable Y.    Step 2    Now here comes the only twist: You start not on the first try (on which it is impossible to duplicate a previous roll), but on the second. So X is one more than Y, hence           f      X        (    t    )    =          f      Y        (    t    −    1    )  , and     E    (    X    )    =    E    (    Y    +    1    )

Consider the experiment of rolling a fair die independently until the same number/face occurs 2 successive times and let X be the trial on which the repeat occurs, e.g. if the rolls are 2,3,4,5,1,2,4,5,5, then X=9. a. find the probability function f(x)=P(X=x). b. compute EX

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