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.
Understanding Statistics: Continuous vs Discrete Distributions
The type of variable comes from the process being studied, not just from the way data are written down. The number of messages received in an hour is a count. It cannot be two and a half messages.
The waiting time for the next message is measured and can be recorded with increasing precision. In real data, continuous measurements are often rounded.
A student may record a height as one hundred sixty eight centimetres, but the person’s actual height was not restricted to whole centimetres. Rounding can make continuous data look like counts, so it is important to think about what was measured before choosing a model.
For a count variable, each possible result has its own probability. A bar chart can show this clearly because the height of each bar represents the chance of that result. There are usually gaps between the bars because values between the counts are not possible.
For example, the number of sixes in ten dice rolls can be zero, one, two, up to ten. Its distribution is shaped by repeated trials. Results near the middle are often more likely than extreme results.
The expected value is the long run average found by weighting every possible value by its probability. This helps in situations such as estimating average numbers of faulty items in a batch.
A smooth curve needs a different interpretation. Its height shows density, which describes how tightly outcomes are packed near a value. Probability comes from the area across a range.
A narrow range in a very dense part of the curve may have more probability than a wider range in a sparse part. Density can even be greater than one without causing a problem, because density is not itself a probability. It has units linked to the measurement unit, such as per second or per centimetre.
An exact measured value has zero probability in a continuous model, yet values extremely close to it can be quite likely. This is why real questions about time, mass, temperature, and speed normally use intervals and tolerances.
Histograms can cause confusion because they use touching bars for continuous measurements. A histogram is not automatically a discrete probability chart. Its bars represent intervals called bins.
With equal width bins, taller bars indicate more observations in that interval. With unequal width bins, the area of each bar must represent frequency or probability, not simply its height. Changing bin widths can make the same data appear to have a different shape.
When learning this topic, pay close attention to the wording. Words such as number of, how many, and count often suggest a discrete variable.
Words such as length, duration, and mass usually suggest a continuous one. Then check whether the question asks for one exact outcome or a range, since that choice determines the method.
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.