Random Variables and Expected Value: Quick Revision

A brief note on Random Variables and Expected Value. Read it before a test. Try each question at the end before you open its answer.

Tables

• Random variable: a number that comes from chance. Discrete ones count, continuous ones measure.
• Distribution: each value $x$ with $P(X = x)$. Each is from $0$ to $1$, and they add up to $1$.
• From a survey: the probability of a value is its count divided by the number asked.

Expected value

• Formula: $E(X) = \mu = \sum x\,P(X = x)$.
• Meaning: the mean in the long run. A simulation of many trials settles near it.
• Totals: the expected value of a total is the total of the expected values.

Spread

• Variance: $\sigma^2 = \sum (x - \mu)^2 P(X = x)$.
• Shortcut: $\sigma^2 = E(X^2) - \mu^2$.
• Standard deviation: $\sigma$, in the unit of $X$. It is the population kind.
• Changing units: adding a constant leaves $\sigma$ alone. Multiplying by $a$ multiplies $\sigma$ by $\lvert a \rvert$.

A worked example

A made-up hockey team scores $0$, $1$, $2$ or $3$ goals in a match with probabilities $0.15$, $0.35$, $0.3$ and $0.2$.

1. Check. $0.15 + 0.35 + 0.3 + 0.2 = 1$.
2. Expected goals. $E(X) = 0.35 + 0.6 + 0.6 = 1.55$ goals.
3. Spread. $E(X^2) = 0.35 + 1.2 + 1.8 = 3.35$, so $\sigma^2 = 3.35 - 1.55^2 = 0.9475$ and $\sigma \approx 0.973$ goals.

Practice with new numbers

A program to try

The program checks the worked example with exact fractions.

Mistakes to avoid

• A plain average in place of a weighted one.
• The most likely value taken as the expected value.
• Stopping at the variance when the standard deviation is asked.
• Trusting a short simulation to match the expected value.

Check yourself

1. $X$ is $0$, $1$ or $2$ with probabilities $0.4$, $0.3$ and $0.3$. Find $E(X)$.

2. Find the variance of the $X$ of question 1.

3. A random variable has $\sigma = 4\ \text{cm}$. What is $\sigma$ in millimetres?

4. A shop expects $2.5$ returns a day. How many should it plan for in $30$ days?

Sources

• OpenStax, Introductory Statistics, Chapter 4, sections on discrete random variables, their distributions, the expected value and the standard deviation.