Name of the formula which links the expectation of a random variable with the summation of the product between conditioned expectation and probability I'd like to know the English name of the following property of the expectation ($X$ random variable; $(A_i)_{i=1}^n$ partition of a set):
\begin{equation}
\mathbb{E}[X] = \sum \limits _{i = 1}^{n} \mathbb{E}[X | A_i] \,\, P(A_i)
\end{equation}
where:
\begin{equation}
\mathbb{E}[X | A_i] = \sum \limits _{k \in \mathbb{R}} k \,\, P(X = k | A_i)
\end{equation}

The first equation is the law of total expectation.  The second equation is the definition of the conditional expected value of a discrete random variable.  (In this latter equation you have used a summation over an uncountable set, which is unusual in probability and statistics; while this is well-defined, it is rather off-putting in this context, and it would be better to take the sum over the support/integers.)
