approximate probability of geometric distribution using CLTFor i ≥ 1, let X i ∼ G 1 / 2 be distributed Geometrically with parameter 1/2. Define Y n = 1 n ∑ i = 1 n ( X i − 2 )Approximate P ( − 1 ≤ Y n ≤ 2 ) with large enough n. Hint, note that Y n is not "properly" normalized.

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approximate probability of geometric distribution using CLTFor   i  ≥  1, let       X    i    ∼      G    1        /    2 be distributed Geometrically with parameter 1/2. Define       Y    n    =      1          n            ∑          i      =      1        n    (      X    i    −  2  )Approximate   P  (  −  1  ≤      Y    n    ≤  2  ) with large enough n. Hint, note that       Y    n   is not &quot;properly&quot; normalized.

kliersel12g · Accepted Answer

Step 1To normalize       Y    n   you need to divide by   σ. Let me explain:Notice that because Xi is distributed geometrically with parameter 1/2, we know that   μ  =      1          1              /            2        =  2 and   σ  =            1      −      p        p    =      2          2      . Furthermore, the normalization of the sum of all       X    i   is                     ∑                  i          =          1                          n                            X        i            −      μ      n              σ              n              =                    ∑                  i          =          1                          n                            X        i            −      2      n              2              n                    /                    2            . This could further more be written as:            2        2        1          n            ∑          i      =      1              n        (      X    i    −  2  )    =      1    σ        1          n            ∑          i      =      1              n        (      X    i    −  2  )Step 2Finally, you just have to use a z-table to get the probabilities such that   P  (            −      1        σ    ≤  Y  ≤      2    σ    )  =  0.6818

For i ge 1, let X_i∼G_1/2 be distributed Geometrically with parameter 1/2. Define Y_n=1/sqrt{n} sum_{i=1}^{n}(X_i-2).

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