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.
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.