Probability distributions describe how the possible outcomes of a random process are spread out. The key difference is whether the variable is discrete, meaning it takes separated countable values, or continuous, meaning it can take any value in an interval. This distinction matters because it changes how probabilities are represented, calculated, and interpreted.
Discrete distributions use probabilities at individual points, while continuous distributions use area under a curve.
Key Facts
- A discrete random variable takes countable values such as 0, 1, 2, 3, ...
- A continuous random variable can take any real value in an interval, such as 1.52 m or 1.521 m.
- For a discrete distribution, probability is found with a probability mass function: P(X = x).
- For a continuous distribution, probability is found with a probability density function: P(a ≤ X ≤ b) = ∫ from a to b f(x) dx.
- For a discrete distribution, total probability satisfies Σ P(X = x) = 1.
- For a continuous distribution, total area satisfies ∫ from -∞ to ∞ f(x) dx = 1, and P(X = exact value) = 0.
Vocabulary
- Random variable
- A random variable is a numerical quantity whose value depends on the outcome of a random process.
- Discrete distribution
- A discrete distribution gives probabilities for countable outcomes, often shown with bars or points.
- Continuous distribution
- A continuous distribution describes probabilities over intervals using area under a smooth density curve.
- Probability mass function
- A probability mass function, or PMF, assigns a probability to each possible value of a discrete random variable.
- Probability density function
- A probability density function, or PDF, gives the density of probability for a continuous random variable, with probabilities found from areas.
Common Mistakes to Avoid
- Treating a PDF value as a probability is wrong because probability in a continuous distribution is area over an interval, not the height of the curve at one point.
- Using P(X = x) for a continuous variable as if it can be positive is wrong because the probability of one exact value in a continuous distribution is 0.
- Forgetting that probabilities must add or integrate to 1 is wrong because every valid probability distribution must include all possible outcomes with total probability 1.
- Choosing a continuous model for count data is wrong when the variable can only take whole-number values, such as number of defects or number of goals.
Practice Questions
- 1 A discrete random variable X has P(X = 0) = 0.20, P(X = 1) = 0.35, and P(X = 2) = 0.25. What must P(X = 3) be if these are the only possible values?
- 2 A continuous random variable has PDF f(x) = 1/4 for 0 ≤ x ≤ 4 and f(x) = 0 otherwise. Find P(1 ≤ X ≤ 3).
- 3 A factory records the number of cracked tiles in each box, while a lab records the exact drying time of paint samples. Identify which variable is discrete and which is continuous, and explain which one would use a PMF and which would use a PDF.