9.1 Discrete random variables

A random experiment produces an outcome; a random variable turns that outcome into a number you can study. It might count successes, record a cost, or give a score. This is the bridge from basic probability to expected value.

This is foundational to intermediate probability. Begin with OpenStax's discrete-distribution lesson. You should know sample spaces and event probabilities first.

A basic idea: one number for each outcome

Let $X$ be the number of heads in two fair tosses. The four equally likely outcomes are $TT,TH,HT,HH$, and $X$ assigns them $0,1,1,2$. Several outcomes can produce the same value. The distribution gathers their probabilities.

A quick tour of properties

For a discrete variable $X$, its probability mass function is $p_X(x)=P(X=x)$. Every value has $0\le p_X(x)\le1$, and the probabilities over all possible values add to $1$. To find $P(X\ge k)$, add the probabilities at values at least $k$. A count always takes nonnegative integer values; a payoff can be negative.

Application 1: count heads

What is the chance of at least one head in two fair tosses?

1. The table gives $P(X=1)=1/2$ and $P(X=2)=1/4$.
2. These possible values cannot occur together, so add them.
3. $P(X\ge1)=1/2+1/4=3/4$.

Application 2: a small reward game

A machine gives $5$ points for a red token, $2$ points for blue, and no reward for green. Their probabilities are $0.1$, $0.3$, and $0.6$. Let $Y$ be the reward.

The probabilities sum to $1$. The chance of a reward of at least $2$ is $0.3+0.1=0.4$. The distribution is about rewards, so its values are $0,2,5$, not the colour names.