A random variable assigns a number to each outcome of a chance process. Its probability distribution lists every possible value and its probability. Edit a distribution, check whether it is valid, and shade the values that make up an event like "at most 2" or "more than 3."
A valid discrete distribution has every probability between 0 and 1, and the probabilities add to exactly 1.
A continuous random variable can take any value in an interval. Its probabilities are areas under a density curve whose total area is 1, so P(X = exactly one value) = 0 and ≤ means the same as <.
A bus arrives at a random time in a 20-minute window, so the wait X is uniform from 0 to 20 minutes. The density is a flat line at height 1/20 = 0.05.
A random variable, like X = number of heads in 3 flips, takes numerical values that depend on the outcome of a chance process. Use capital letters for the variable and lowercase for a specific value.
A discrete random variable takes a fixed set of separate values, often counts like 0, 1, 2, 3. Its distribution is a table or a probability histogram, and you find probabilities by adding bars.
A continuous random variable, like a waiting time, can take any value in a range. Probabilities are areas under a density curve, and any single exact value has probability 0.
Every probability must be between 0 and 1, and all the probabilities must add to 1. A missing probability is often found by subtracting the others from 1.
For a discrete variable, "at least 3" includes 3, but "more than 3" does not. Translate the words into ≤, <, ≥, or > before adding probabilities.
The cumulative probability P(X ≤ x) adds every probability up to and including x. It rises from the first value to 1 at the largest value.
For a discrete random variable with values 0, 1, 2, …, these phrases mean different sets of values.
| Phrase | Symbol | Values included (k = 3) | Shortcut |
|---|---|---|---|
| Exactly 3 | P(X = 3) | 3 | Read the table |
| At most 3, no more than 3 | P(X ≤ 3) | 0, 1, 2, 3 | Cumulative probability at 3 |
| Fewer than 3, less than 3 | P(X < 3) | 0, 1, 2 | Same as P(X ≤ 2) |
| At least 3, 3 or more | P(X ≥ 3) | 3, 4, 5, … | 1 − P(X ≤ 2) |
| More than 3, greater than 3 | P(X > 3) | 4, 5, … | 1 − P(X ≤ 3) |
| Between 2 and 4 (inclusive) | P(2 ≤ X ≤ 4) | 2, 3, 4 | P(X ≤ 4) − P(X ≤ 1) |
| For a continuous X | P(X < 3) = P(X ≤ 3) | An interval, not a list | Area under the density curve |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 2.8. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.