Revise Discrete random variables and expected value for IB Mathematics: Analysis and Approaches (4.4)

A random variable is a variable whose value is a numerical outcome of a random experiment. We write it with a capital letter such as $X$, and use the matching lowercase letter $x$ for a particular value it can take. A random variable is discrete when the values it can take form a finite or countably infinite list of separate numbers — the kind of thing you count. The score on a die, the number of sixes in five rolls, and the number of faulty bulbs in a box are all discrete. A continuous variable, by contrast, could be any value in an interval, like a length or a time; those are not the subject of this topic.

A probability distribution for a discrete random variable $X$ pairs each possible value of $X$ with its probability. It is almost always displayed as a table, with the values $x$ across the top and the probabilities $P(X=x)$ beneath. Whatever else a table shows, two conditions must hold for it to be a genuine probability distribution.

The expected value of a discrete random variable is the number you would get by averaging the variable over a very large number of repetitions of the experiment. It is a weighted average of the possible values, where each value is weighted by how likely it is. Because it is the mean of the distribution it is also written $\mu$.

$E(X) = \mu = \sum x\,P(X=x)$

In words: multiply every outcome $x$ by its probability $P(X=x)$, then add all those products together. Two features of $E(X)$ trip students up. First, it is a long-run average, so it need not equal any value the variable can actually take — the expected score on a fair die is $3.5$, which no single roll can produce. Second, it is not the same as the most likely outcome (the mode); a lopsided distribution can have its mean well away from its most probable value.

Expected value is the backbone of decisions under uncertainty — in insurance, finance and gambling. A game is fair when a player's expected gain is zero: over the long run they neither win nor lose. If the variable you are working with is the player's profit (winnings minus stake), fairness means $E(\text{profit})=0$. If instead you are working with the gross winnings, the equivalent condition is that the expected winnings equal the stake, $E(\text{winnings})=\text{stake}$. A positive expected profit favours the player; a negative one favours the house, which is why commercial games are always tilted so the operator's expected profit is positive.

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A), where an A-mark is dependent on the corresponding M-mark being earned. Follow-through (FT) means a correct value carried on from an earlier slip still earns the later marks, and ISW ('ignore subsequent working') means once you have written a correct answer, extra scribble does not lose it. The examples above already carry the [M]/[A] tags line by line; the one below shows the full protocol on a classic 5-mark distribution question, and then spells out exactly how each mark is awarded.

The expected value is the first of the summary numbers that describe a distribution; the variance and standard deviation, which measure spread, are built on the very same weighted-sum idea. The habit you have practised here — tabulate the outcomes, enforce $\sum P(X=x)=1$, then weight and sum — reappears directly in the binomial distribution, where the probabilities come from a formula rather than a table, and again in continuous distributions such as the normal, where the sum becomes an integral. Master the routine now and each of those topics becomes a variation on a method you already own.