Mathematics · University
Continuous distributions and densities · revision notes
A density is integrated over a set; its value at a point is not the point probability.
Key ideas and conditions
Density versus probability
A density is integrated over a set; its value at a point is not the point probability.
Normalization
A density must be nonnegative and integrate to one.
Expected value
E[g(X)]=∫g(x)f(x)dx when the integral exists.
Cumulative distribution
F(x)=P(X≤x); differentiating an absolutely continuous CDF recovers a density almost everywhere.
Explain without looking
- Which proposed function cannot be a probability density?
- Find the constant c making f(x)=c x² on [0,1] a density.
Further reading and review references
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