A probability density function, or PDF, describes how likely different values of a continuous variable are. It is used when outcomes can take any value in an interval, such as height, time, mass, or temperature. Unlike a bar graph for discrete outcomes, a PDF is a smooth curve, and probability is found from the area under the curve.
This idea matters because many real measurements are modeled with continuous distributions, especially the normal distribution.
Understanding Statistics: Probability Density Functions
The height of a curve is called density, not probability. This distinction prevents a common mistake. A tall, narrow part of a curve can have less probability than a lower, wider part because width matters.
Imagine comparing two regions of a graph. One is high but very thin. The other is lower but stretches across a large range of values.
The second region may contain more observations. Density has units that depend on the measurement.
If time is measured in seconds, density is measured per second. When density is multiplied by a width measured in seconds, the units cancel and leave a probability.
Probabilities for ranges are often found with a cumulative distribution function. This function records the area from the far left of the graph up to a chosen value. To find the chance that a measurement falls between two values, subtract the earlier cumulative area from the later one.
In practice, software, tables, or calculators do this area calculation. Students should still connect the numerical result to the graph.
A result near zero means the selected interval covers little of the distribution. A result near one means it covers nearly all of it.
The normal model is useful because many measurements cluster around a typical value. Its center is set by the mean. Its spread is set by the standard deviation.
A larger standard deviation produces a wider and flatter curve. A smaller standard deviation produces a narrower and taller curve, while keeping the same total area. This explains why a high peak does not automatically mean a larger chance for every interval.
Real data are not always normal. Test scores can be uneven, waiting times are often skewed, and incomes can have a long right tail. A model should be chosen by looking at the data rather than by assuming a familiar curve fits.
Measurements add another practical issue. A recorded value such as 170 centimetres usually represents a small interval, not a perfectly exact height. The person may be anywhere within the rounding range.
This is why continuous models work naturally with measurements, even when a data table lists separate rounded numbers. When reading a density graph, check the horizontal scale, the interval being discussed, and whether the curve is a reasonable model for the situation.
Do not treat the vertical height as a percentage. Focus on the area over the stated range, because that area represents the fraction of observations expected there over many repeated measurements.
Key Facts
- For a continuous random variable X with density f(x), P(a <= X <= b) = integral from a to b of f(x) dx.
- The total area under any valid PDF is 1: integral from -infinity to infinity of f(x) dx = 1.
- A PDF is never negative: f(x) >= 0 for all x.
- The probability at one exact point is zero: P(X = a) = 0.
- For a uniform density on [a, b], f(x) = 1/(b - a) for a <= x <= b.
- For a normal distribution, f(x) = 1/(sigma sqrt(2 pi)) e^(-(x - mu)^2/(2 sigma^2)).
Vocabulary
- Probability density function
- A function that describes the relative likelihood of values of a continuous random variable, where probability is measured by area under the curve.
- Continuous random variable
- A variable that can take any value within an interval, including decimals and fractions.
- Area under the curve
- The region between a PDF and the x-axis over an interval, representing the probability that the variable falls in that interval.
- Normal distribution
- A symmetric bell-shaped probability distribution described by its mean and standard deviation.
- Cumulative probability
- The probability that a random variable is less than or equal to a chosen value.
Common Mistakes to Avoid
- Reading the height of the PDF as the probability. The probability is the area over an interval, not the curve's y-value at one point.
- Assigning a positive probability to one exact value. For a continuous random variable, P(X = a) = 0 because a single point has no width and no area.
- Forgetting that the total area must equal 1. A function cannot be a valid PDF if its total area over all possible x-values is not exactly 1.
- Using interval endpoints incorrectly. For continuous variables, P(a < X < b), P(a <= X <= b), P(a < X <= b), and P(a <= X < b) all have the same value because single endpoints have probability zero.
Practice Questions
- 1 A continuous random variable has PDF f(x) = 1/5 for 0 <= x <= 5 and f(x) = 0 otherwise. Find P(1 <= X <= 4).
- 2 A PDF is f(x) = cx for 0 <= x <= 2 and f(x) = 0 otherwise. Find c, then find P(0 <= X <= 1).
- 3 A student says the most likely exact value in a normal distribution has a large probability because the curve is highest there. Explain what is wrong and how probability should be found.