Random variables and probability tables

What you will be able to do

• say what a random variable is, and tell a discrete one from a continuous one;
• build a probability table from counts, and check that it is a real distribution;
• read probabilities such as $P(X \ge 2)$ from a table or a bar chart.

A question with a number for an answer

A made-up class survey asks $40$ students one question: how many brothers and sisters do you have? Now pick one student at random. Before you ask, you cannot know the answer. It could be $0$, $1$, $2$ or $3$.

A number like this, which comes from a chance process, is a random variable. We give it a capital letter, such as $X$. A small letter, such as $x$, stands for one of its values. So "$X = 2$" means "the student picked has two brothers or sisters".

A random variable that counts things, like this one, is discrete: its values are separate numbers. A random variable that measures, such as a height or a waiting time, is continuous: it can take any value in a range. This course is about discrete ones. The last course of the chapter meets a continuous one, the normal distribution.

A probability table

Here are the survey counts. A student picked at random has the answer $1$ with probability $\dfrac{18}{40} = 0.45$, because $18$ of the $40$ students said $1$.

The last row is the probability distribution of $X$. It gives every value with its probability. Every distribution obeys two rules.

1. Each probability is between $0$ and $1$.
2. The probabilities add up to $1$, because $X$ always takes exactly one of its values.

Here $0.2 + 0.45 + 0.25 + 0.1 = 1$, so the table passes.

A bar chart shows the same table as a picture. The tallest bar is the most likely value. The shape tells you at a glance that most students have one brother or sister, and few have three.

You should find that the bars always share out a total of $1$, whatever the counts. Double every count and nothing changes, because each probability is a share of the class.

Reading a table

Many questions ask for more than one value at once. Add the probabilities of the values you want.

• $P(X \ge 2) = P(X = 2) + P(X = 3) = 0.25 + 0.1 = 0.35$.
• $P(X \le 1) = 0.2 + 0.45 = 0.65$. This is also $1 - 0.35$, the complement.

A distribution can also come from equally likely outcomes. Think of a family with two children, and suppose each child is equally likely to be a girl or a boy. The four orders GG, GB, BG and BB are equally likely. So the number of girls has $P(0) = \dfrac{1}{4}$, $P(1) = \dfrac{2}{4}$ and $P(2) = \dfrac{1}{4}$.

Try it in Python

The program counts the answers of the made-up survey, turns each count into a probability and draws the bar chart.

Your turn

The usual mistakes

• A table that does not add up to $1$. Check the total every time. A missing value or a typing slip shows up at once.
• Dividing by the wrong total. A probability from a survey is a count divided by the number of people asked.
• Mixing up $X$ and $x$. $X$ is the random variable. $x$ is one of its values.
• Forgetting the end values. "At least $2$" includes $2$. "More than $2$" does not.

Remember these

• A random variable is a number that comes from chance.
• Its distribution gives each value $x$ with $P(X = x)$. Each probability is from $0$ to $1$, and they add up to $1$.
• Add the probabilities of the values you want. Use the complement when it is shorter.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Which of these are discrete and which are continuous: the number of goals in a match, and the time a runner takes for $100$ m?

2. A table has $P(X = 0) = 0.1$, $P(X = 1) = 0.3$ and $P(X = 3) = 0.2$. The only other value is $2$. Find $P(X = 2)$.

3. A student writes a table with probabilities $0.5$, $0.4$ and $0.3$. Can it be a probability distribution?

4. A spinner has four equal parts labelled $1$, $1$, $2$ and $5$. Let $X$ be the number it lands on. Write the distribution of $X$.

5. Use the survey table. What is the probability that a student picked at random has fewer than two brothers and sisters?

References

• OpenStax, Introductory Statistics, Chapter 4, section on probability distribution functions for a discrete random variable.