I chose a random number from 1 to 6. Afterwards, I roll a die till I get a result that is even or higher than my chosen number. What is the E(x) of the number of times I throw the die?So I thought it's Geometric distribution will "success" where "success" is to get my number. So first the probability of choosing a random number in the die is 1/6. Now I can't configure what is the probability to get something even or higher? Since I chose the number randomly. and afterthat, I just need to divide 1 by the probability I get (like the geometric E(x) formula)?

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I chose a random number from 1 to 6. Afterwards, I roll a die till I get a result that is even or higher than my chosen number. What is the E(x) of the number of times I throw the die?So I thought it&#039;s Geometric distribution will &quot;success&quot; where &quot;success&quot; is to get my number. So first the probability of choosing a random number in the die is 1/6. Now I can&#039;t configure what is the probability to get something even or higher? Since I chose the number randomly. and afterthat, I just need to divide 1 by the probability I get (like the geometric E(x) formula)?

akademiks1989rz · Accepted Answer

Step 1Let X be uniformly distributed on {1,…,6} and let N be the number of times that you need to roll the die. Note that   N  ∣  X  =  x is geometrically distributed with probability mass function   P  (  N  =  k  ∣  X  =  x  )  =  (  1  −  p      )          k      −      1        p    (      k    ≥    1    ) where   p  =            6      −      x      +      1        6  .Step 2The law of total expectation yields that  E  N  =  E  (  E  N  ∣  X  )  =  E      6          6      −      X      +      1        =  6  E      1          6      −      X      +      1        =  6  ×      1    6        (    1    +          1      2        ⋯          1      6        )    .So   E  N  =      ∑          k      =      1        6        1    k

I chose a random number from 1 to 6. Afterwards, I roll a die till I get a result that is even or higher than my chosen number. What is the E(x) of the number of times I throw the die?

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