We're given a random sample of X 1 , X 2 , . . . . , X n from a distribution with a pdf of f ( x ; θ ) = θ x ( 1 − θ ) , x = 0 , 1 , 2 , . . . . and 0 &lt; θ &lt; 1Show that that T n = ∑ i = 0 n X i is a complete sufficient statistic for θ.

Question

We&#039;re given a random sample of       X          1        ,      X          2        ,  .  .  .  .  ,      X          n       from a distribution with a pdf of  f  (  x  ;  θ  )  =      θ    x    (  1  −  θ  )  ,  x  =  0  ,  1  ,  2  ,  .  .  .  . and   0  &amp;lt;  θ  &amp;lt;  1Show that that   T  n  =      ∑          i      =      0        n        X          i       is a complete sufficient statistic for   θ.

Houston Ochoa · Accepted Answer

X    i    is a geometric random variable. Thus       T    n   is the number of tails that appear before the   nth heads, when repeatedly flipping a coin that has a   θ chance of being tails. You can show that the PMF is  P  (      T    n    =  k  )  =                    (                            n        +        k        −        1            k                      )                  θ    k    (  1  −  θ      )    n

We're given a random sample of X_1,X_2,....,X_n from a distribution with a pdf of f(x;theta)=theta^x(1−theta),x=0,1,2,.... and 0<theta<1 Show that that Tn= sum_i=0^n X_i is a complete sufficient statistic for theta.

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