Discrete random variables and their distributions

What a random variable is

• A random variable assigns a number to each outcome of a random situation.
• It is written with a capital letter ($X$), and the values it can take with a lower-case letter ($x$). $P(X = x)$ is the probability that the variable takes the value $x$.
• Discrete means the variable takes separate, countable values — usually whole numbers. There is nothing between 2 and 3.
• Continuous means the variable can take any value in a range, including fractions of whatever precision you like — height, time, mass.

• The distinction decides which tools apply. Discrete variables have a probability for each value; continuous variables have zero probability at any exact value, and probabilities come from areas over intervals.

The distribution of a discrete random variable

• The probability distribution lists every value with its probability, as a table, a graph or a formula.
• Two conditions define a valid distribution:

$$0 \leqslant P(X = x) \leqslant 1 \qquad \text{and} \qquad \sum P(X = x) = 1$$

• Check the sum first, every time. It is how missing values are found and how invalid tables are exposed.

The discrete uniform distribution

• The simplest model: $n$ outcomes, all equally likely.

$$P(X = x) = \frac{1}{n} \quad \text{for each of the } n \text{ values}$$

• Use it when the situation has a fixed set of outcomes with no reason to favour any — a fair die, a randomly selected day of the week, a raffle ticket drawn from a drum.
• Do not use it just because a variable has several possible values. Equal likelihood is a strong assumption and usually requires a symmetric physical mechanism.

Cumulative probabilities

• The cumulative probability $P(X \leqslant x)$ adds all probabilities up to and including $x$.
• It is what your calculator's "cdf" functions give, and most exam questions need it:

• This table is worth memorising. For a discrete variable, the difference between $<$ and $\leqslant$ is a whole bar of probability, and mixing them up is the commonest calculator error in this standard.

Building a distribution from a situation

• List every possible value.
• Find the probability of each, using a tree, a table or systematic counting.
• Check the probabilities sum to 1.
• Present it as a table, and graph it if asked.