# 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 ðŸ˜€