Continuous Random Variables and Density Functions
For a continuous variable, probability is area under a density, so any single value has probability zero. The cumulative function and the density are linked by calculus in both directions.
What a learner can do afterwards
- Verify that a function is a valid density and find a probability from it
- Differentiate a cumulative function to recover the density
- Explain why P(X = a) is zero and what that does not mean
1 · Read
A continuous variable can land anywhere in an interval, so you never ask for the chance of one exact value: P(X = a) = 0. Instead you draw a density curve and read chances as areas under it. A valid density never dips below zero and encloses total area exactly 1. Check X uniform on [0, 4] with height 1/4: positive everywhere, with width 4 times height 1/4 = 1. Both rules pass.
Uniform chances are rectangle areas. P(1 < X < 3) on [0, 4] is width 2 times height 1/4 = 0.5. Shaped curves work the same way: f(x) = x over 8 on [0, 4] is valid, and P(X < 2) is the triangle with base 2 and height 2 over 8, so half times 2 times 0.25 = 0.25. Buses spread evenly over 0 to 20 minutes give P(wait between 5 and 10) = 5 times 1 over 20 = 0.25.
Height is not chance. Mia reads f(2) = 1/4 as P(X = 2) = 1/4, but the chance at a single point is zero. Zero does not mean impossible: the bus can arrive at exactly 5 minutes, yet that instant has zero width and zero area. Only intervals carry chance, which is what the zero does and does not mean.
The CDF and the density link both ways: differentiate F to recover f, since F = x over 4 on [0, 4] gives f = 1 over 4. Integrate f over an interval for its chance. Whenever a story says values spread evenly, draw the rectangle first.
Densities turn chance into area: nonnegative curve, total area one, points zero.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.