# Sum of probabilities of all samples gives the total volume?

I am working on a churn problem (binary classification - whether a customer will churn or not). Now using logistic regression, I get the probability whether the customer will churn or not. Can I add up these probabilities (rows are distinct per user) to get the total volume of predicted churn? (expected value of the total number of users that will churn)?

How would I prove this statistically? $$E(X) = \sum_{x=1}^{n} x_i P(x_i)$$ $$where \ X:churn\ volume, x_i = 1 \ (1 \ for \ every \ user), P(x_i):probability \ from \ the \ classification \ model$$

1. Does this look right? Or what might be the right working?
2. Does the equations change when there is a class imbalance. Please correct me or point me in the right direction

You have $$n$$ customers, for each customer $$i$$ you have the value $$c_i$$ telling us whether the customer churned ($$c_i=1$$) or not ($$c_i=0$$), and you have the belonging probabilities $$p_i := p(c_i=1)$$ and $$1-p_i = p(c_i = 0)$$. As always with binary random variables, the expectation $$E[c_i]$$ is $$p_i$$: $$E[c_i] = 0\cdot (1-p_i) + 1\cdot p_i = p_i.$$
Now, you want to compute the expectation of the sum $$\sum_{i=1}^n c_i$$. But the expectation operation is linear, thus: $$E[\sum_{i=1}^n c_i] = \sum_{i=1}^n E[c_i] = \sum_{i=1}^n p_i.$$