Diyelim ki iki bağımsız Bernoulli rasgele değişken olan B e r ( θ 1 )
Bunu nasıl kanıtlarız ( ˉ X 1 - ˉ X 2 ) - ( θ 1 - θ 2 )√θ 1 ( 1 - θ 1 )n 1 +θ2(1-θ2)n, 2 d→N(0 , 1 )
N 1 ≠ n 2 olduğunu varsayın
Diyelim ki iki bağımsız Bernoulli rasgele değişken olan B e r ( θ 1 )
Bunu nasıl kanıtlarız ( ˉ X 1 - ˉ X 2 ) - ( θ 1 - θ 2 )√θ 1 ( 1 - θ 1 )n 1 +θ2(1-θ2)n, 2 d→N(0 , 1 )
N 1 ≠ n 2 olduğunu varsayın
Yanıtlar:
Put
a=√θ1(1−θ1)√n1
Since A
This proof is incomplete. Here we need some estimates for uniform convergence of characteristic functions. However in the case under consideration we can do explicit calculations. Put p=θ1, m=n1
Note that similar calculations may be done for arbitrary (not necessarily Bernoulli) distributions with finite second moments, using the expansion of characteristic function in terms of the first two moments.
Proving your statement is equivalent to proving the (Levy-Lindenberg) Central Limit Theorem which states
If {Zi}ni=1
Here ˉZ=∑iZi/n
Then it is easy to see that if we put
Zi=X1i−X2i
E(Zi)=θ1−θ2=μ
and
V(Zi)=θ1(1−θ1)+θ2(1−θ2)=σ2
(There's a last passage, and you have to adjust this a bit for the general case where n1≠n2