Introduction to Random Variables and Probability Distributions

A food truck owner can't know how many drinks the next customer will order, but over thousands of orders a pattern is steady: 30% of orders have no drink, 45% have one, 15% have two, and 10% have three. A list like that, every possible number paired with its probability, is a probability distribution, and it's the object the rest of Unit 2 works with.

Random variables

A random variable is a numerical value determined by the outcome of a random process. Before the process runs, you don't know its value; afterward, the outcome fixes it. Random variables get capital letters, like $X$, and a particular value gets the lowercase letter, like $x$. So "$X = 2$" is the event that the random variable comes out to 2, and $P(X = 2)$ is its probability.

The definition has two parts that matter. The value must be a number: "the color of the next car" comes from a random process but isn't a random variable, while "the number of red cars among the next 10" is. And the number must come from chance: the number of days in March is always 31, so it isn't random.

A discrete random variable has values you can list with gaps between them, usually counts: the number of drinks in an order, the number of heads in 5 flips, the number of defective chips in a box. A random variable that can take any value in an interval, like the time until the next customer arrives, is continuous; topic 2.11 handles that kind. This topic is about discrete random variables.

Probability distributions

The probability distribution of a discrete random variable gives the probability of each of its possible values. Let $X$ = the number of drinks in a randomly selected order from the food truck:

Two requirements make a table like this a valid distribution, and both come straight from topic 2.4:

Here $0.30 + 0.45 + 0.15 + 0.10 = 1$, so the table is valid. The second requirement also lets you find one missing probability: if a table left out $P(X = 3)$, it would have to be $1 - (0.30 + 0.45 + 0.15) = 0.10$.

To find the probability of an event about $X$, add the probabilities of the values in it, because the values are mutually exclusive. "At least two drinks" is $X \ge 2$:

With a discrete variable, the inequality sign changes the answer. $P(X > 2)$ leaves out the value 2, so it's only $0.10$.

Three ways to show a distribution

A table, like the one above, lists each value and its probability.

A graph, usually a probability histogram, puts the values on the horizontal axis and draws a bar over each one whose height is its probability.

Shape, center and spread are read from it the same way as for a histogram of data (topic 1.6): this distribution peaks at 1 drink and is skewed right, with a tail toward 3.

A function gives the probability as a formula in $x$. You'll see one in the next section.

Building a distribution from probability rules

When the random process is simple enough to list its outcomes, you can work the distribution out exactly. Roll two fair four-sided dice (faces 1 to 4) and let $X$ = the larger of the two numbers (if they match, that number). There are $4 \times 4 = 16$ equally likely outcomes. Sort them by the value of $X$:

• $X = 1$: only (1, 1). Probability $\frac{1}{16}$.
• $X = 2$: (1, 2), (2, 1), (2, 2). Probability $\frac{3}{16}$.
• $X = 3$: (1, 3), (2, 3), (3, 3), (3, 1), (3, 2). Probability $\frac{5}{16}$.
• $X = 4$: the 7 outcomes with at least one 4. Probability $\frac{7}{16}$.

The probabilities sum to $\frac{16}{16} = 1$. The pattern has a reason: $X \le k$ means both dice are at most $k$, which happens in $k^2$ outcomes, so $X = k$ exactly accounts for $k^2 - (k-1)^2 = 2k - 1$ outcomes. That gives the distribution as a function:

The steps are always the same: list the sample space, find the value of $X$ for each outcome, and add the probabilities of the outcomes that give each value. When outcomes aren't equally likely, use the multiplication and addition rules from topics 2.6 and 2.7 to get each probability.

Estimating a distribution by simulation

Some processes are tedious to work out exactly. A simulation (topic 2.3) estimates the distribution instead: run the process many times with random numbers, record the value of $X$ in each trial, and use the relative frequency of each value as its estimated probability. More trials give estimates that tend to be closer to the true probabilities.

Cumulative distributions

A cumulative probability distribution gives, for each value $x$, the probability that $X$ is less than or equal to $x$: $P(X \le x)$. Build it by keeping a running total down the table. For the food truck:

The cumulative column never decreases and ends at exactly 1, because every value of $X$ is at most the largest one. It answers "at most" questions in one look: $P(X \le 1) = 0.75$, so three quarters of orders have at most one drink. With the complement rule it answers "more than" questions too, $P(X > 1) = 1 - P(X \le 1) = 0.25$, and differences of cumulative values give ranges: $P(1 \le X \le 2) = P(X \le 2) - P(X \le 0) = 0.90 - 0.30 = 0.60$.

For the dice maximum, the cumulative distribution is also a function: $P(X \le x) = \frac{x^2}{16}$ for $x = 1, 2, 3, 4$, which is the $k^2$ count from before.

Worked examples

Where students lose points