Is this equality involving max true?

Question

Suppose I have a sequence of two non-negative R.V. $\{X\}_{i=1}^n$ and $\{Y\}_{i=1}^n$, and suppose $X_i$ are i.i.d. Can I conclude that $$\max_{1\leq i\leq n}\{X_i+Y_i\}=\max_{1\leq i\leq n}\{X_i\}+\max_{1\leq i\leq n}\{Y_i\}\,\,\,?$$ The doubts arises because intuitively if $X_i+Y_i$ are departure and arrival times of some costumer then splitting the max may give the departure time of customer $x$ and the arrival time of costumer $y$ with $x\neq y$. But this would not make much sense as $\max\{X_i+Y_i\}$ should give the longest arrival and departure of $1$ costumer, not $2$. Would the fact that they are i.i.d maybe play a role here?

Answer 1

It's not true for real numbers. Try $n=2$. If $(X_1,Y_1,X_2,Y_2)=(0,1,1,0)$ then $\max(X_1+Y_1, X_2+Y_2) = \max(1,1) = 1$ while $\max(X_1, X_2)=1$ and $\max(Y_1,Y_2)=1$.

The above example shows that the proposed identity can't hold when we have random variables, either. Just take $X_1$, $Y_1$, $X_2$, $Y_2$ to be IID Bernoulli random variables (i.e., they take values only zero and one); the probability that the proposed identity fails will be at least the probability that $(X_1,Y_1,X_2,Y_2)=(0,1,1,0)$.