Does this function computing a conditional expectation have a standard name?

Question

I am thinking this function may have a standard name that I should call it. If not, I'd be interested in what you mathematicians would call it...

$expected(R_{\lnot e1}, P_{e1}, R_{e1 \lnot e2}, P_{e2}, R_{e1 \land e2})$ $= (1 - P_{e1}) * R_{\lnot e1} + P_{e1} * ((1 - P_{e2}) * R_{e1 \lnot e2} + P_{e2} * R_{e1 \land e2})$

where Pe2 is probability that event2 happens given event1 has already happened, where Pe1 is the probability that event1 happens, where the R's are the resultant values in the three cases (event1 didn't happen, event1 happened but event 2 did not, or event1 and event2 both happened). For example: cost is 0 if tile never breaks off, cost is 10 if tile breaks off but does not shatter, cost is 40 if tile breaks off and shatters.

Is there a standard or common name for that? Or what's the best name for that? (I realize that it is "an expected value"... but there's lots of ways to compute expected values... this is a specific sequence of events type situation.)

Answer 1

Your question is a bit unclear, but if I understand you correctly, you're just talking about the expected value of a random variable $R$. To check my understanding, here's what I'm assuming:

• We check if an event $e_1$ has occurred, which it does with probability $P_{e_1}$.
  • If it hasn't, then $R$ gets the value $R_{\neg e_1}$.
  • If, instead, $e_1$ has occurred, then we check if another event $e_2$ has occurred, which it does with probability $P_{e_2}$.
    • If it hasn't, then $R$ gets the value $R_{e_1, \neg e_2}$.
    • If it has, then $R$ gets the value $R_{e_1, e_2}$.

In that case, we would say that

$$ E(R) = (1-P_{e_1})R_{\neg e_1} + P_{e_1}\left[(1-P_{e_2})R_{e_1, \neg e_2} + P_{e_2}R_{e_1, e_2}\right] $$

There isn't a special name for this that I can think of, other than expected value.

Answer 2

If $R$ represents a reward, then your formula gives $E(R)$ which is usually called... expected reward (or expected value of the reward, sometimes and somewhat improperly mean reward).