Let T be a complete suﬃcient real-valued statistic for the parameter θ and S be another real-valued statistic whose distribution function F does not depend on the parameter θ. Show that P ( S ≤ s ∣ T ) − F ( s ) equals zero almost surely under each P θ and for each real s. Conclude from this that S and T are independent.

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Let   T be a complete suﬃcient real-valued statistic for the parameter   θ and   S be another real-valued statistic whose distribution function   F does not depend on the parameter   θ. Show that   P  (  S  ≤  s  ∣  T  )  −  F  (  s  ) equals zero almost surely under each       P    θ   and for each real   s. Conclude from this that   S and   T are independent.

cismadmec · Accepted Answer

The general term of an arithmetic sequence is given by the formula:            a      n        =    a    +    d          (      n      -      1      )      where a is the initial term and d the common difference.Given             a      8        =    26   and             a      12        =    42  , we find:      16    =    42    -    26                      16              =          a      12        -          a      8                          16              =          (      a      +      11      d      )        -          (      a      +      7      d      )                          16              =    4    d  Hence d=4Then:      26    =          a      8        =    a    +    7    d    =    a    +    28  Hence a=−2So the first five terms of the sequence are:−2,2,6,10,14

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