Expectation of a discrete random variable

If X is a random variable having probability distribution (probability mass function) p(x), the expected value of X is:
\[E[x] = \sum\limits_{x:p(x) > 0} {xp(x)}\]
1. E[X] when X is a Bernoulli random variable with parameter p:
\[E[x] = p\]
2. E[X] when X is a binomial random variable with parameter n and p:
\[E[x] = np\]
3. E[X] when X is a Poisson random variable with parameter$\lambda$:
\[E[x] = \lambda\]
These expectation is only for selected discrete random variables.


Ref:Sheldon M. Ross, Introduction to probability Models, ch 2.4