Fair die roll 8 times, number of of perfect squaresYou have a fair die you roll 8 times. Let X be the number of times you roll a perfect square. What is E ( X 2 )?I tried doing summation from 1 to 8 of ( n / 8 ) ⋅ 2.5 ) + ( 8 − n ) / 8 ⋅ ( 3.5 ) and divide by 8 but that was wrong. G

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Fair die roll 8 times, number of of perfect squaresYou have a fair die you roll 8 times. Let X be the number of times you roll a perfect square. What is   E  (      X    2    )?I tried doing summation from 1 to 8 of   (  n      /    8  )  ⋅  2.5  )  +  (  8  −  n  )      /    8  ⋅  (  3.5  ) and divide by 8 but that was wrong. G

Shyla Maldonado · Accepted Answer

Step 1It is easy to see that   X  ∼  Bin  ⁡  (  8  ,  1      /    3  ). That is, X follows a binomial distribution with 8 trials and probability of success 1/3. Thus   E  (  X  )  =  8      /    3.To compute   E      (          X      2        )   without resorting to the sum definition, we can be clever: Recall that   Var  ⁡  (  X  )  =  E      (          X      2        )    −  [  E  (  X  )      ]    2  . The variance of X is  Var  ⁡  (  X  )  =  n  p  (  1  −  p  )  =      16    9  Step 2So   E      (          X      2        )    =      16    9    +      64    9    =      80    9  which is twice the value you computed in the comments.

varsa1m · Accepted Answer

Step 1Doing summation is not wrong, if you know how to calculate probabilities from binomial distribution,Step 2                    E                  (                      X            2                    )                                    =                  ∑                      x            =            0                                8                          P                  (          X          =          x          )                ⋅                  x          2                                                  =                  ∑                      x            =            0                    8                                                    (                                            8            x                                              )                                                            (                          1              3                        )                    x                                      (                          2              3                        )                                8            −            x                                    x          2                                                  =                  1                      3            8                                    ∑                      x            =            0                    8                                                    (                                            8            x                                              )                                                2                      8            −            x                                    x          2                                                  =                              1            ⋅            256            ⋅            0            +            8            ⋅            128            ⋅            1            +            28            ⋅            64            ⋅            4            +            56            ⋅            32            ⋅            9                        +            70            ⋅            16            ⋅            16            +            56            ⋅            8            ⋅            25            +            28            ⋅            4            ⋅            36            +            8            ⋅            2            ⋅            49            +            1            ⋅            1            ⋅            64                                3            8                                                            =                  58320                      3            8                                                            =                  80          9

Fair die roll 8 times, number of of perfect squares. You have a fair die you roll 8 times. Let X be the number of times you roll a perfect square. What is E(X^2)?

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