I'm having trouble understanding one of the steps of the Wald test used in two-sample binomial proportions. More specifically, when $X_1, ..., X_{n_1}$ are iid Binomial$(1, p_1$), $Y_1, ..., Y_{n_2}$ are iid Binomial$(1, p_2$), with these two samples mutually independent, and the hypothesis being $H_0: p_1=p_2$ versus $H_1:p_1\neq p_2$

Supposing $n_1/n \to\lambda_1 \in(0,1)$, and $n_2/n \to\lambda_2 \in(0,1)$, as $n=n_1+n_2\to\infty$, using the central limit theorem and Slutsky's theorem the following apparently can be found: $\sqrt n[(\hat p_1-\hat p_2)-(p_1-p_2)]$ converges in distribution to $N(0,\frac{1}{\lambda_1}p_1(1-p_1)+\frac{1}{\lambda_2}p_2(1-p_2))$

Could someone give me some insight or a basic proofing of how the final term can be found? Thank you in advance.



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