I've tried to gather information about these three topics i'm about to show but i couldn't find. I will display them in a question for better understanding and context of the situation.
For a and b real numbers, define:
$a∨b := max ${a, b} and $a∧b := min${a,b}
Let X and Y be such that $E(X∨Y) = 20$, and E(X∧Y) = 10. Get E|X-Y|.
SO, what's the ∨, ∧ and := symbols meaning that appear throughout the problem? If anyone could solve for better understanding and briefly talk about the min max part i'd appreciate.
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$\begingroup$All that is occurring here is that the author is defining a notation, which is what the := sign is telling you. In other words, the up carrot is a relation between two numbers that returns after operation the number which is smaller of $a$ and $b$. Similarly the down carrot is a relation between two numbers that returns the number which is larger of $a$ and $b$.
So if we let $a=2, b=3$, then $2 \lor 3 = \max{\{2,3\}}=3$. And $2 \land 3 = \min{\{2,3\}}=2$
I can honestly say that I've never seen that notation myself during my probability studies. But I suppose since the author is defining it in his own way, using := to tell us so, then I suppose that is how the author wants his students to learn it. I've always seen a new random variable defined just as say, let $Z=\max{\{X,Y\}}$ for r.v's $X,Y$.
$\endgroup$ 2 $\begingroup$Note $$|X-Y| = \max\{X,Y\} - \min\{X,Y\}$$ and then take expectations and substitute.
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