I read a statement in a book and I can't realize what exactly it means, and why it is true.
a set of random variables are given. in general, the expected value of the maximum of random variables is not equal to the maximum of the expected values of these variables.
I would like to know what exactly it means and why it is true.
thanks
Take two independent coins $c_1,c_2$ which have equal probability of heads or tails.
Assign a value of $1$ for a head and $0$ for a tail.
Then $E c_1 = E c_2 = {1 \over 2}$ and $\max (E c_1, E c_2) = {1 \over 2}$.
Now consider $\max(c_1,c_2)$, that is, toss both coins and take the 'best' result. Then $E [\max(c_1,c_2)] = {3 \over 4}$ (just look at all 4 possibilities).
Hence $\max (E c_1, E c_2) \neq E [\max(c_1,c_2)]$.
Addendum: Note that in the first case, heads and tails are equally likely so $E c_1 = {1 \over 2} 0 + {1 \over 2} 1 = {1 \over 2}$.
In the second case, each pair of possibilities is equally likely so \begin{eqnarray} E [\max(c_1,c_2) &=& {1 \over4} \max(0,0) + {1 \over4} \max(0,1) +{1 \over4} \max(1,0) +{1 \over4} \max(1,1)\\ &=& {1 \over4} 0 + {1 \over4} 1 +{1 \over4} 1 +{1 \over4} 1 \\ &=& {3 \over 4} \end{eqnarray}