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 the density or its cumulative function
- Differentiate a cumulative function to recover the density
- Explain why P(X = a) is zero and what that does not mean
1 · Read
For a variable with a continuous density, P(X = a) = 0 at every single value. Discrete models assign probabilities to countable values, which need not be whole numbers. A density gives probabilities as areas under its curve. It must be nonnegative and have total area 1. For X uniform on [0, 5], the density is 1/5 there and 0 elsewhere. Its area is 5 × 1/5 = 1, so both rules pass. For a right-tail probability, subtract the area to its left from 1.
Uniform chances are rectangle areas. P(1 < X < 4) on [0, 5] is width 3 times height 1/5 = 0.6. Shaped curves work the same way. Say f(x) = kx on [0, 2] is a density. The area is a triangle with base 2 and height 2k, so 2k = 1 and k = 1/2. Then P(X < 1) is a triangle with base 1 and height 1/2, so half times 1 times 1/2 = 0.25, and P(X > 1) = 0.75.
Height is not chance. Leo reads f(2) = 1/5 as P(X = 2) = 1/5, but the chance at a single point is zero. In a continuous model for a timer, 3 minutes can be a possible outcome even though P(X = 3) = 0: the point has zero width and zero area. An interval's probability is its area under the density. Adding or removing an endpoint changes no probability because a single point contributes zero.
F(x) = P(X ≤ x) is the cumulative function. Where F is differentiable, the density is its slope. For F(x) = x / 5 on [0, 5], the density inside the interval is 1 / 5. For F(x) = x squared / 25 on [0, 5], differentiate x squared to get 2x, then divide by 25: f(x) = 2x / 25 inside the interval. Interval probabilities are areas under f: P(a < X < b) = F(b) - F(a). The fundamental theorem of calculus links area and slope.
A density turns chance into area: the curve is never negative, the total area is 1, and a single point has area zero.
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