Discrete Random Variables

Starting point: Years 9 and 10

Relative frequency

If you repeat an experiment many times,

$$\text{relative frequency}=\frac{\text{number of times it happened}}{\text{number of trials}}$$

This gets closer to the true probability as the number of trials grows.

Expected number of outcomes

$$\text{expected number}=\text{number of trials}\times \text{probability}$$

Roll a die 120 times and you expect about $120\times \frac{1}{6}=20$ sixes.

Probability tables

A table can list every outcome and its probability. The probabilities must add to 1.

Quick summary

• Relative frequency approaches probability over many trials. Expected number $=n\times p$.
• A probability distribution lists each value with its probability. They add to 1.
• $E(X)=\sum xP(X=x)$ is the long-run average. A fair game has $E(X)=0$.
• $\text{Var}(X)=E(X^2)-{\mu }^2$ and $\sigma =\sqrt{\text{Var}(X)}$.
• $E(aX+b)=aE(X)+b$ and $\text{Var}(aX+b)=a^2\text{Var}(X)$.

Quiz