Q.Which of the following is a discrete random variable?
I. The number of cars crossing a particular signal in a day.
II. The number of customers in a queue to buy train tickets at a moment.
III. The time taken to complete a telephone call.

A random variable $X$ is a function defined on a sample space $S$ into the real numbers $\mathbb R$, such that the inverse image of every point, subset or interval of $\mathbb R$ is an event in $S$ (Definition 11.1). It turns the outcomes of a random experiment — which need not themselves be numbers, e.g. a coin's head/tail — into numbers we can compute with. Capital letters $X,Y,Z$ denote the random variable itself; small letters $x,y,z$ denote its possible values. If $x$ is a possible value, its inverse image $X^{-1}(x)=\{\omega\in S:X(\omega)=x\}$ is always an event of $S$, so it has a probability — this is what makes "$P(X=x)$" meaningful.

Two flavours are studied:

• Discrete random variable (Definition 11.2): its range is countable — finite, or a sequence $x_1,x_2,x_3,\dots$ — with every value carrying positive probability and the whole range's probabilities summing to $1$. Used for counting a quantity: number of heads, number of defectives, a sum of dice faces, a winning amount taking finitely many values. A discrete random variable can even live on a continuous sample space (e.g. a step function of $\omega\in[0,20]$ taking only two values is still discrete, because its range, not its domain, is what is tested for countability). …