A random variable, x, is a numerical description of the outcome of an experiment. Here, an experiment is in reference to ones in probability, not in the sciences. An experiment in probability is defined as any process that generates some outcomes. For example, tossing a coin is an experiment as it can result in either heads or tails. A random variable assigns values to the outcomes of an experiment, such as 1 to heads and 0 to tails. There are two kinds of random variables: discrete and continuous. A discrete random variable takes values such as 0, 1, 2 and so on while a continuous random variable can take any value inside some interval.
A discrete probability distribution describes how probabilities are distributed over values of a discrete random variable. Discrete probability distributions are defined by the probability function, f(x), which provides probabilities for each value of the random variable. There are two requirements for the probability function. The first requirement is that each f(x) is greater than or equal to zero. The second requirement is that the sum of the probabilities f(x) are equal to one.
The expected value and variance of a discrete random variable are measures of central location and variability. Just as the mean of a sample or population of data is a measure of central location, the expected value, or mean, of a random variable is a measure of central location for the random variable. Similarly, just as the variance of a sample or population of data is a measure of variability for the data, the variance of a random variable is a measure of variability for the random variable. The standard deviation is the square root of the variance. It's also a measure of variability for the random variable but is in the same units as the random variable.
The binomoial probability distribution is the probability distribution associated with a binomial experiment. A binomial experiment is an experiment that satisfies four properties. The first property is that the experiment consists of a sequence of n identical trials. The second property is that there are two possible outcomes possible on each trial: "success" and "failure". The third property is that the probability of success, p, does not change from trial to trial. Similarly, the probability of failure, 1-p, does not change from trial to trial. The fourth property is that the trial are independent. Then if the random variable x is defined as the number of success in n trials, x has a binomial probability distribution.