Expected value uses probability to tell us what outcomes to expect in the long run. It represents the average outcome or the long-term average that can be anticipated from a probability distribution. Informally, the expected value is the arithmetic mean of the possible values a random variable can take, weighted by the probability of those outcomes. The expected value of a random variable $X$ is denoted by $E[X]$.
For discrete distributions
$$ E[X] = \sum_{i=1}^{N}x_i \cdot P(x_i) $$
For example, given a fair coin flip where landing heads $H$ results in winning $10 and landing tails $T$ results in winning $0, the expected value of winnings can be calculated as
$$ E[X] = 10 \times \frac{1}{2} + 0 \times \frac{1}{2} = 5 $$
For continuous distributions
Now consider a random variable X which has a probability density function given by a function f on the real number line. This means that the probability of X taking on a value in any given open interval is given by the integral of f over that interval. The expectation of X is then given by the integral
$$ E[X] = \int_{-\infty}^{\infty} xf(x)dx $$
where $f(x)$ is the PDF.
For uniform distributions
The expected value for a uniform distribution in the range $[a,b]$ is given by
$$ E[X] = \frac{a+b}{2} $$