Random variables
When a chance process produces a number, how do you describe every value it might take, where it centers and how far it spreads?
Where you are 9 inside
What this area is about
Often what matters about an outcome is a number attached to it: shots of espresso in a drink, calls until a pledge, dollars won. That number is a random variable X. This area lists its values with their probabilities, then summarizes them with a mean and a standard deviation, and names two patterns that turn up again and again.
How these fit together
Describing one variable. A probability distribution table gives each value of X and its probability, and cumulative probability answers questions about a range.
Center and spread. Expected value \mu_X = \sum x \, P(x) is the long-run average. The variance and standard deviation measure spread.
Results for changed and combined variables. Changing every value by a rule a + bX gives the new mean a + b\mu_X and standard deviation |b|\sigma_X: adding moves only the center, multiplying rescales both. Variances add for a sum or difference, when the variables are independent.
Two named distributions. The binomial distribution counts successes in a fixed number n of independent trials with the same chance p. The geometric distribution counts independent trials, each with the same chance p, until the first success, and its mean is 1/p.
A path through it
The distribution table first, then cumulative probability, expected value and variance. Next, adding a constant, then variances adding. The binomial distribution needs the table and the variance; the geometric formula follows it, then the geometric mean.
Finish here and density curves carry these ideas to continuous variables.