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I am a math instructor self studying for the actuarial exam and I am trying to understand the following notation that I have encountered today.

$$E[X \land d]$$

The explanation in the book told me that this means

$$E[\min\{X,d\}]$$

which is another unfamiliar notation to me.

Just guessing from what I have learned I want to say that this is related to reimbursements with deductibles with $X$ being the loss which I learned it as

$$E[Y]$$ while $$Y = \begin{cases} 0, & x<d\\ x-d, & x\ge d \end{cases}$$

Am I in the right ball park or does it mean something completely different? It would be great if you could guide me to where I can learn about this a bit more because I do not even know how it is read.

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    $\begingroup$ You are a math instructor and you do not know what min means? Anyway, it would help if you gave more context, e.g., you write $d$ but do not say what it is. $\endgroup$ – KCd Aug 31 '14 at 1:16
  • $\begingroup$ d stands for the deductible. it's just a constant. anyway, like KCd and even you yourself said, min stands for min. i.e. the minimum. what is the expected value of the minimum of X and d? $\endgroup$ – Tyler Aug 31 '14 at 1:21
  • $\begingroup$ Haha, it's interesting how you would think that all math instructors know what you already do. Anyhow, to me $\land$ means the intersection of a logic statement and I was not sure that it was related to the minimum value of X and d. It is just my first time hearing that it also meant the minimum value. Thanks for the help, anyway. $\endgroup$ – hyg17 Sep 1 '14 at 22:01
  • $\begingroup$ (fun fact) as for where the notation comes from... $\wedge$ and $\vee$ are the "meet" and "join" of a lattice. when you're looking at a lattice based on order (like x < y), it makes sense to consider the max as the join and the min as the meet $\endgroup$ – Tyler Sep 2 '14 at 13:26
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This notation isn't particular to actuaries (see here). $\wedge$ means the minimum. Dually, $\vee$ would mean maximum.

$E(f(x))=\int f(x)p(x) \mathrm{d}x$ by definition.

Thus you get

$E(\min(X,d)) = \int \min(x,d)p(x) \mathrm{d}x$

$\min(X,d)$ means "take $X$ if $X < d$, otherwise take $d$."

Another notation you may come across (particularly for max) is $(S_T - K)^+ = \max(S_T-K,0)$ which is the payoff of a call option.

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  • $\begingroup$ Thank you for your precise help. It makes it much clearer for me. $\endgroup$ – hyg17 Sep 1 '14 at 22:03

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