A probability density function, or PDF, describes how probability is spread across the possible values of a continuous random variable. Instead of assigning probability to one exact value, it assigns density along the number line. This matters because measurements like height, time, mass, and voltage are often modeled as continuous quantities.
Calculus connects the curve to probability through area under the graph.
Key Facts
- For a valid PDF, f(x) >= 0 for all x.
- The total area under a PDF is 1: integral from -infinity to infinity of f(x) dx = 1.
- Probability over an interval is area: P(a <= X <= b) = integral from a to b of f(x) dx.
- For a continuous random variable, P(X = a) = 0 because a single point has zero width.
- The mean or expected value is E[X] = integral from -infinity to infinity of x f(x) dx.
- If f(x) = 0 outside an interval, only integrate over the interval where the density is nonzero.
Vocabulary
- Probability density function
- A function f(x) that gives probability density for a continuous random variable and whose total area is 1.
- Continuous random variable
- A variable that can take any value in an interval rather than only separated values.
- Area under the curve
- The integral of a density function over an interval, which represents probability.
- Expected value
- The long run average value of a continuous random variable, found by integrating x f(x).
- Normalization
- The requirement that the total integral of a probability density function equals 1.
Common Mistakes to Avoid
- Treating f(a) as P(X = a), which is wrong because density is not the same as probability. For a continuous variable, the probability at one exact point is 0.
- Forgetting that total area must equal 1, which makes the function invalid as a probability density function. Always check that the integral over the whole domain is 1.
- Using height instead of area to compare probabilities, which is wrong because interval width also matters. Probability is found by integrating over an interval.
- Integrating over the wrong bounds, which gives the probability of the wrong event. Match the lower and upper limits to the exact interval stated in the problem.
Practice Questions
- 1 Let f(x) = 2x for 0 <= x <= 1 and f(x) = 0 otherwise. Verify that f(x) is a valid probability density function by computing the total integral.
- 2 For f(x) = 2x on 0 <= x <= 1, find P(0.25 <= X <= 0.75).
- 3 A probability density curve is tall and narrow near x = 2 but shorter and spread out from x = 5 to x = 9. Explain why the wider region might represent a larger probability even though the curve is lower there.