Year 12 / Continuous random variables and normal distributions
Probability is an area.
A probability density must be non-negative and integrate to one over its support. The density height is not itself a probability; a height can exceed one. Probability over an interval is the area under the density.
For a continuous variable, the probability at one exact point is zero. The cumulative distribution F(x) gives P(X ≤ x) and is non-decreasing. Expectation integrates x ; variance can be found using E() − μ².
INTERACTIVE MODEL
Try it. Watch it change.
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Density is zero outside [0, 1]. A shaded interval gives probability; a single point has probability zero.
Explore: Slide the upper bound and compare the shaded area with the cumulative probability.
Interval probability
The density is = for 0 ≤ x ≤ 1. Find P(X ≤ 0.5).
- Integrate from 0 to 0.5: []₀^0.5.
- The probability is 0.25.
Assumed knowledge
Definite integrals and probability rules.
Learning checkpoints & source
YOUR LEARNING CHECKPOINT- Validate a density and calculate interval probabilities.
- Find expectation and use a cumulative distribution.
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