Suppose that X 1 , … , X n are n i.i.d. random variables from the Poisson distribution truncated on the left at 0.. Find the UMVUE of P ( X 1 = 1 ) ..

Question

Suppose that       X    1    ,  …  ,      X    n   are n i.i.d. random variables from the Poisson distribution truncated on the left at   0.. Find the UMVUE of   P  (      X    1    =  1  )  ..

ehedem26 · Accepted Answer

To add to the final line in Michael Hardy&#039;s answer, we have that  E  (  T  ∣      X    1    +  ⋯  +      X    n    )  =  Pr  (      X    1    =  1  )  ⋅            Pr      (              X        2            +      ⋯      +              X        n            =      x      −      1      )              Pr      (              X        1            +      ⋯      +              X        n            =      x      )      Now given       X    1    +      X    2    ⋯  +      X    n    =  x in the denominator, we can say that       X    2    +  ⋯  +      X    n    =  x  −      X    1  So in the numerator, we obtain       X    2    +  ⋯  +      X    n    =  x  −      X    1    =  x  −  1 , giving       X    1    =  1.Hence  E  (  T  ∣      X    1    +  ⋯  +      X    n    )  =  Pr  (      X    1    =  1  )  ⋅  Pr  (      X    1    =  1  )Hence  E  (  T  ∣      X    1    +  ⋯  +      X    n    )  =  Pr  (      X    1    =  1  )  ⋅  Pr  (      X    1    =  1  )and this is easily obtainable by substituting in   x  =  1 into   Pr  (  X  =  x  )  =            λ      x              x      !      (              e        λ            −      1      )

Suppose that X_1,…,X_n are n i.i.d. random variables from the Poisson distribution truncated on the left at 0. Find the UMVUE of P(X_1=1).

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