Continuous distributions
The density and the cumulative distribution function, probability seen as an area, then the continuous uniform, exponential (with its memorylessness) and normal distributions.
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A few cards from the deck
Probability density function: what is it?
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A non-negative function of total area 1 whose area under the curve, between a and b, gives the probability that the variable falls between a and b.
Its value at a point is not a probability and may exceed 1; only areas are. Hence P(X = a) = 0 for any value a.
A variable has constant density c on [0; 4], zero elsewhere. What is c?
0.25140.5
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0.25The total area must be 1: c × 4 = 1, so c = 1/4. A density is not a probability, but here it stays below 1; on [0; 0.5] it would be 2.
For a variable with a density, the probability of a single value is …: only … under the density are probabilities.
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For a variable with a density, the probability of a single value is 0: only areas under the density are probabilities.
A point has no width, hence no area. That does not make the value impossible, but it means P(X < 3) = P(X ≤ 3).
Cumulative distribution function: what is it?
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The function F(x) = P(X ≤ x): the area under the density to the left of x. It rises from 0 to 1.
It gives P(a < X ≤ b) = F(b) − F(a) directly and is used to read medians and quantiles.