random variable
A random variable is a way of attaching a number to the uncertain outcome of an experiment, so that we can do arithmetic with chance. The outcome of a coin toss is 'heads' or 'tails' — words, not numbers. But if we say 'heads = 1, tails = 0', we now have a number that varies randomly from toss to toss. That number is a random variable, usually written with a capital letter like X.
More carefully, a random variable is a rule that assigns a number to every outcome in the sample space. The number of claims on a policy this year is a random variable taking values 0, 1, 2, ...; the size of the next fire loss is a random variable taking values across a continuous range. Once outcomes are numbers, we can ask for their average (expectation), their spread (variance), and the probability that they land in any range. The random variable is the bridge from the messy real world of events to the clean machinery of distributions.
Nearly everything an actuary computes is the expected value or the variability of some random variable. The present value of a life insurance benefit, the aggregate losses of a portfolio, the time until a policyholder dies — each is modeled as a random variable, and the premium or reserve is built from its expectation and its spread. Learning to see a real-world quantity as a random variable, then choosing a distribution for it, is the central habit of the trade.
Let X = the number of claims a policy produces in a year. X is a random variable; it might be 0 most years, occasionally 1, rarely more. The premium is built largely from E[X], the average value of X.
Naming the random variable is the first step in pricing any risk.
A random variable is the rule (the function), not the single number you happen to observe — that observed number is called a 'realization' of the variable.