Continuing Alignment After Text

Question

I currently have the following output:

I want to align the equals signs even after the bit of text. Is there a nice way to do this besides having one big align block and putting the text inside it? (The code is below if needed.)

We have
\begin{align}
\mathbb{E}X &= \sum_{k=0}^\infty k\mathbb{P}(k)\
&=\sum_{k=0}^\infty k\frac{\lambda}{k!}e^{-\lambda}\
&=\lambda e^{\lambda}\sum_{k=0}^\infty k\frac{1}{k!}\
&=\lambda e^{\lambda}\sum_{k=1}^\infty (k-1)\frac{1}{(k-1)!}.
\end{align}
Performing the change of variable $j=k-1$ we then get
\begin{align}
&=\lambda e^{\lambda}\sum_{j=0}^\infty j\frac{1}{j!}\
&=\lambda e^{-\lambda}e^\lambda\
&=\lambda.
\end{align}

Edit: Also please ignore the typos in the equation itself, I just hastily copied down an example :)

Answer 1

I suppose that this is what you are after:

\documentclass{report}
\usepackage{amsmath}
\usepackage{amsfonts}
\begin{document}
\begin{align*}
\mathbb{E}X &= \sum_{k=0}^\infty k\mathbb{P}(k)\\
&=\sum_{k=0}^\infty k\frac{\lambda}{k!}e^{-\lambda}\\
&=\lambda e^{\lambda}\sum_{k=0}^\infty k\frac{1}{k!}\\
&=\lambda e^{\lambda}\sum_{k=1}^\infty (k-1)\frac{1}{(k-1)!}.\\
\intertext{Performing the change of variable $j=k-1$ we then get}
&=\lambda e^{\lambda}\sum_{j=0}^\infty j\frac{1}{j!}\\
&=\lambda e^{-\lambda}e^\lambda\\
&=\lambda.
\end{align*}
\end{document}

The output will be: