I'm given a random variable X n that is wrt the following measure: lim n → ∞ # { X n ∈ [ a , b ] } n = ∫ a b sin 2 ⁡ θ d θ .with an error term O ( log ⁡ n / n ).Now I'm trying to find the distribution of cos 2 ⁡ X n . The text I'm reading says this is ∫ a b cos 2 ⁡ θ d θ, which I'm already not sure of, and I also want to figure out what the error term of this is.

Question

I&#039;m given a random variable       X    n   that is wrt the following measure:      lim          n      →      ∞                  #      {              X        n            ∈      [      a      ,      b      ]      }        n    =      ∫    a    b        sin    2    ⁡  θ  d  θ  .with an error term   O  (  log  ⁡  n      /    n  ).Now I&#039;m trying to find the distribution of       cos    2    ⁡      X    n  . The text I&#039;m reading says this is       ∫    a    b        cos    2    ⁡  θ  d  θ, which I&#039;m already not sure of, and I also want to figure out what the error term of this is.

gozaderaradiox5 · Accepted Answer

(1)                              lim                      n            →            ∞                                                #            {                          X              n                        ∈            [            a            ,            b            ]            }                    n                =        k                  ∫          a          b                          sin          2                ⁡        θ        d        θ                          w        h        e        r        e                           k          =                      2            π                              Please note that I have added a factor   k  =            2      π      . See later.First of all, as the values taken by the LHS of (1) are between 0 and 1, the same must be true for the RHS of (1).This is why I have added factor   k in order that      2    π        ∫          0              π            sin    2    ⁡  θ  d  θ  =  1.Let       f    X   and       F    X   be the pdf and cdf resp. of random variable   X.The LHS of (1) is plainly       ∫    a    b        f    X    (  x  )  d  x under the classical &quot;subjectivist&quot; assumptions of probability (I think that the error term   O  (  log  ⁡  (  n  )      /    n  ) comes from a theorem of De Finetti, not sure, but we don&#039;t need it ; I would even say that I find its presence a little strange).Therefore, if identity (1) is valid for any   a  ,  b, they define a same measure ; therefore:      f    X    (  x  )  =  k      sin    2    ⁡  (  x  )  =  k      1    2    (  1  −  cos  ⁡  (  2  x  )  )  =      1    π    (  1  −  cos  ⁡  (  2  x  )  )giving, by integration:      F    X    (  x  )  =      {                            0                          for                     x          &amp;lt;          0                                                  1            π                    (                      1            2                    x          −                      1            4                    sin          ⁡          (          2          x          )          )                          for                     x          ∈          [          0          ,          π          ]                                      1                          for                     x          &amp;gt;          π                        Let   Y  :=      cos    2    ⁡  X.Let us compute the cdf of   Y which can take values   y  ∈  [  0  ,  1  ].      F    Y    (  y  )  =  P  (  Y  &amp;lt;  y  )  =  P  (  −      y    &amp;lt;  cos  ⁡  X  &amp;lt;      y    )  =  2  P  (  cos  ⁡  X  &amp;lt;      y    )(last equality by symmetry of the pdf of   X wrt   π      /    2)      F    Y    (  y  )  =  2  P  (  X  &amp;lt;                                          cos                          −              1                                ⁡          (                      y                    )                ⏟              c    )  =  2      F    X    (  c  )                    (2)                              F          Y                (        y        )        =        2                  1          π                [                  1          2                c        −                  1          4                sin        ⁡        (        2        c        )        ]            It remains to express (2) in a simpler form, and finally to take the derivative of cdf       F    Y   with respect to   y in order to get the pdf       f    Y  .

I'm given a random variable X n </msub> that is wrt the following measure:

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