How to write a covariance Could somebody kindly explain how you can express something as a covariance?
A paper I am reading contains this term:
$$\sum_in_i(p_i-p)(b_a+\sum_t\sum_jr^t\frac {n_j} {n_i}b_c)$$
The author then says that the term "can be expressed using a covariance" as follows:
$$N\mathbb C[p_i,(b_a+\sum_tr^t\sum_j\frac {n_j} {n_i} b_c)] $$
N is the sum of all n and $p$ is the average of all $p_i$.
It's a fascinating paper, but I've hit a block based on my neophyte maths. Please could somebody put me out of my misery and explain how such a covariance can be found? 
Would love to hear an explanation so I'll be able to understand in the future!

Covariance of  two random variables $X$ and $Y$ is defined as:
$$
\mathbb C[X,Y] = E_{x,y}[XY] - E_x[X]E_y[Y]
$$
where $E_x[X]$ represents the expected value of r.v $X$. 
$E[p_i]=p$
$$\begin{align}N\mathbb C[p_i,(b_a+\sum_tr^t\sum_j\frac {n_j} {n_i} b_c)] & =NE_i[p_i(b_a+\sum_tr^t\sum_j\frac {n_j} {n_i} b_c)]\\ & -NE_i[p_i]E_i[b_a+\sum_tr^t\sum_j\frac {n_j} {n_i} b_c] \\
&= N\sum_i\Big(p_i(b_a+\sum_tr^t\sum_j\frac {n_j} {n_i} b_c)\times\frac{n_i}{N}\Big)\\ &-N\times p\times \sum_i\Big(b_a+\sum_tr^t\sum_j\frac {n_j} {n_i} b_c \Big)\times \frac{n_i}{N}\\
&= \sum_i n_ip_i(b_a+\sum_tr^t\sum_j\frac {n_j} {n_i} b_c)-\sum_i n_ip(b_a+\sum_tr^t\sum_j\frac {n_j} {n_i} b_c)
\end{align}$$
