probability - What is the average of rolling two dice and only taking the value of the higher dice roll?

What is the average result of rolling two dice, and only taking the value of the higher dice roll?

To make sure the situation I am asking about is clear, here is an example:
I roll two dice and one comes up as a four and the other a six, the result would just be six.

Would the average dice roll be the same or higher than just rolling one dice?

Answer

The number of ways to roll a number $x$ under your definition would be $2(x-1) + 1$.

Therefore the expected value would be

$$E[X] = \sum_{x=1}^6\frac{2(x-1)+1}{36}x = \frac{1}{36}\sum_{x=1}^6(2x^2 - x) = \frac{161}{36} \approx 4.47$$
So the average is considerably higher than the average of a single die, being $3.5$.