probability – What is $mathbb{E}[(1 – (1 – p)^{X})(n-X-1)]$ as $n rightarrow infty$?


Let $X sim text{Bin}(n-1,p)$ and $p = frac{lambda}{n}$. What is $mathbb{E}((1 – (1 – p)^{X})(n-X-1))$ as $n rightarrow infty$ ?

Here is my attempt:

begin{align}
mathbb{E}((1 – (1 – p)^{X})(n-X-1)) &= mathbb{E}((n-1) – X – (n-1)(1 – p)^{X} + X(1 – p)^{X})\
&= (n-1) -mathbb{E}(X) – (n-1)mathbb{E}((1 – p)^{X}) + mathbb{E}(X(1 – p)^{X}).
end{align}

begin{align}
mathbb{E}(X) &= (n-1)p\
mathbb{E}((1 – p)^{X}) &= mathbb{E}(e^{ln{(1 – p)^{X}}}) = mathbb{E}(e^{Xln{(1-p)}}) = (1 – p + pe^{ln{(1-p)}})^{n-1} hspace{15mm} (text{by momemnt generating function})\
&= ((1+p)(1-p))^{n-1}\
mathbb{E}(X(1 – p)^{X}) &= sum_{k=0}^{n-1}k(1-p)^{k}{n-1 choose k}p^{k}(1-p)^{(n-1)-k} = sum_{k=1}^{n-1}k(1-p)^{k}{n-1 choose k}p^{k}(1-p)^{(n-1)-k} \
&=(n-1)psum_{ell=0}^{n-2}(1-p)^{ell+1}{n-2 choose ell}p^{k-1}(1-p)^{(n-2)-ell} = (n-1)p(1-p)mathbb{E}((1 – p)^{Y})\
&= (n-1)p(1-p)((1+p)(1-p))^{n-2}, hspace{5mm} text{since } Y sim text{Bin}(n-2,p).
end{align}

Now putting it together, I believe, we obtain:

begin{align}
mathbb{E}((1 – (1 – p)^{X})(n-X-1)) = (n-1)left((1 – p) – ((1+p)(1-p))^{n-1} + p(1-p)((1+p)(1-p))^{n-2}right).
end{align}

Focusing on the term in the square brackets. As $n rightarrow infty$,
begin{align}
(1-p) &= 1-tfrac{lambda}{n} rightarrow 1 \
((1+p)(1-p))^{n-1} &= (1+tfrac{lambda}{n})^{n-1}(1-tfrac{lambda}{n})^{n-1} rightarrow e^{lambda}e^{-lambda} = 1 \
((1+p)(1-p))^{n-2} &= (1+tfrac{lambda}{n})^{n-2}(1-tfrac{lambda}{n})^{n-2} rightarrow 1 \
p(1 -p) &= tfrac{lambda}{n}(1 – tfrac{lambda}{n}) rightarrow 0 \
end{align}

So the term in the square brackets converges toward zero. Moreover, $(n-1) rightarrow infty$. This leaves me with the question where the whole term converges toward?

Important: according to my lecture notes the expectation should converge toward $lambda^2$. The exercise is related to the relationship between a offspring process and the Erdos-Renyi random graph.

Also, this is my first question asked on stack exchanged. I really hope I followed the guidelines of the platform correctly. If not, my apologies. Please let me know how to improve my question 😀