Define discrete random variables and construct and verify discrete probability distributions, including finding unknown probabilities

What this dot point is asking

SCSA Unit 3 introduces discrete random variables as the foundation for the binomial distribution and all of probability in the course. This dot point asks you to define such variables and build and verify their probability distributions, examined in both sections of the WACE written examination.

Discrete random variables

A random variable assigns a number to each outcome of a random process. A discrete random variable takes values that can be listed separately, such as $0,1,2,3$, typically counts. This contrasts with a continuous variable, which takes any value in an interval.

The second condition reflects certainty: one of the listed outcomes must occur, so the probabilities exhaust all possibilities.

Finding an unknown probability

The most common SCSA task is a distribution with one probability written as an unknown. Because the probabilities must sum to $1$, you set up an equation and solve.

Probabilities of events

Once the distribution is known, the probability of a compound event is the sum of the probabilities of the values it includes. For the table above, $P(X\ge 3)=P(X=3)+P(X=4)=0.4+0.2=0.6$, and $P(X<3)=1-0.6=0.4$ by the complement.

The discrete uniform distribution

When all $n$ outcomes are equally likely, each has probability $\dfrac{1}{n}$; this is the discrete uniform distribution. A fair die gives $P(X=x)=\dfrac{1}{6}$ for $x=1,\dots,6$. The uniform case is the simplest distribution and a useful check that probabilities sum to $1$.

Distributions given by a rule

Sometimes the probabilities are not listed in a table but defined by a formula such as $P(X=x)=cx$ or $P(X=x)=c(x+1)$ for stated values of $x$. The method is the same: write the sum of all the probabilities, set it equal to $1$, and solve for the constant.

Cumulative distribution

The cumulative probability $P(X\le x)$ adds all probabilities up to and including $x$, increasing from $0$ to $1$ as $x$ runs through the values. It is the discrete analogue of the area under a density curve and is the quantity the calculator returns for at-most events. Reading off $P(X\le x)$ lets you answer at-least and between questions by subtraction.