How Do You Calculate Expected Value for Complex Probability Density Functions?

[hellosonic]

So for general expected value, I know how to do it. Multiply probability of X to the value of X and add them up. For density function such as

f(x) = 2x 0<x<1 and 0 otherwise. I take find the integral. I'm just confused when it comes to something like

f(x) = (x - 8) 8 < x < 9; (10 -x ) 9 < x < 10; 0 otherwise. I hope someone can help me clear this up lolz. Thanks veryyyyyy much

 

[sylar]

Here since this is continuous random variable, we can define f(9)=1. Then looking at the graph of x-f(x), we see that the expected value is the sum of the areas under the curve. Hence, for this question the expected value is ((10-8)*1)/2=1, which proves that f is a probability function.