# How to read Exp(z|y) in statistics?

I am calculating Customer Lifetime Value and have the following expression:

Customer Lifetime Value = Exp( visits | lifetime) * avg(transaction size)


In context of statistics, what does Exp(z | y) mean?

What is Exp in this context? (exponential, e^(z | y)..?)

What is | in this context? (z*y, z given y..?)

Thanks.

• CLV probably is a random variable, so on the right side of equation you should have a 'recipe' for random variable, and avg is a real number. That being said, Exp(visits|lifetime) could be a conditional expectation. See wikipedia for conditional expectation. Commented Feb 5, 2017 at 11:05
• Thank you. So E(z | y) is more correct? Commented Feb 5, 2017 at 11:10
• I don't know how about you, but I'm used to writing random variables with capitals, so it would be rather E(Z|Y). Small letters 'z' and 'y' usually mean observed value of random variable Z and Y, respectively. Commented Feb 5, 2017 at 11:34
• There are two different things: for a real integrale random variable $X$ and another random variable there is the conditional expectation $E[X|Y]$, see en.m.wikipedia.org/wiki/Conditional_expectation. This is yet another random variable. Then there is the factorization of that, denoted by $E[X|Y=y]$, that is, a function from the space Y maps to to the reals. This quantity is so important in ML because given features X and a real valued outcome Y, one can show that y->E[X|Y=y] is the best regression (with lowest RMSE). They mean E[Visits|Lifetime=lifetime]. Commented Jun 27, 2019 at 6:13

## 1 Answer

Yes, in this case Exp is meant to be $\mathit{E}$, the expectation of number of visits given the lifetime of customer.