A die is thrown at random. What is the expectation and variance

The expectation is the mean.

Add up all the possible outcomes and multiply by the probability for each.

In this case, the outcomes are 1,2,3,4,5,6 and the probability for each is 1/6

(1+2+3+4+5+6)\frac{1}{6}

The variance is \frac{\sum(x-\text{mean})^{2}}{n-1}

Square the difference between each outcome and the mean, then divide by the number of possible outcomes less 1.

You should get the same as the mean.