Random Variable vs Probability Distribution

A random variable turns the outcomes of a random process into numbers you can average and analyze. For example, if $X$ is the number of heads in two coin flips, then $X$ can equal 0, 1, or 2, each with its own probability. Random variables are discrete when their values are countable and continuous when they can take any value in an interval. Its long-run average is the expected value $E(X) = \sum x_i \, p_i$ (E of X, each value times its probability, summed).

A probability distribution shows how the total probability of $1$ is shared among a random variable's possible values. For example, rolling a fair six-sided die gives each value from 1 to 6 a probability of $\frac{1}{6}$. For a discrete variable the probabilities must satisfy $0 \le p_i \le 1$ and $\sum p_i = 1$ (each probability is between 0 and 1, and they add to 1). Continuous variables use a density curve instead, where probability is the area under the curve.