A random variable is a rule that assigns a number to each outcome of a random process. This idea lets us use algebra, graphs, and formulas to describe uncertainty. The most important first split is between discrete random variables and continuous random variables.
Knowing the difference helps you choose the right probability model and avoid common mistakes with probabilities.
Key Facts
- A random variable X assigns a numerical value to each outcome of a random experiment.
- Discrete random variables have countable possible values, such as 0, 1, 2, 3.
- Continuous random variables can take any value in an interval, such as any height between 150 cm and 190 cm.
- For a discrete random variable, probabilities are assigned to exact values: P(X = x).
- For a continuous random variable, probability is area under a density curve: P(a < X < b) = ∫ from a to b f(x) dx.
- For any continuous random variable, P(X = exact value) = 0, even though intervals can have positive probability.
Vocabulary
- Random Variable
- A random variable is a function that assigns a numerical value to each possible outcome of a random process.
- Discrete Random Variable
- A discrete random variable has possible values that can be counted one by one.
- Continuous Random Variable
- A continuous random variable can take any value within an interval or range.
- Probability Mass Function
- A probability mass function gives the probability of each exact value of a discrete random variable.
- Probability Density Function
- A probability density function describes how probability is spread over intervals for a continuous random variable.
Common Mistakes to Avoid
- Treating a continuous variable like it has probability at one exact point is wrong because P(X = a) = 0 for continuous random variables.
- Calling any variable with decimals continuous is wrong because a variable can use decimals but still have a limited or countable set of possible values.
- Forgetting that discrete probabilities must add to 1 is wrong because the total probability over all possible values must equal 1.
- Interpreting the height of a density curve as probability is wrong because probability for a continuous variable is area under the curve, not the curve height alone.
Practice Questions
- 1 A coin is flipped 4 times, and X is the number of heads. Is X discrete or continuous, and what are the possible values of X?
- 2 A discrete random variable has P(X = 0) = 0.20, P(X = 1) = 0.35, and P(X = 2) = 0.45. Find P(X ≥ 1).
- 3 Classify each variable as discrete or continuous and explain your reasoning: the number of texts sent in a day, the time it takes to run a mile, and the temperature in a classroom.