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Using the PDF and CDF

The PDF and CDF are two views of the same distribution, linked by calculus. Knowing which one a question wants is usually the difference between a one-line answer and a mess.

The PDF and the CDF
F(x)=∫−∞xf(t) dtf(x)=ddxF(x)F(x) = \int_{-\infty}^{x} f(t)\,dt \qquad f(x) = \frac{d}{dx}F(x)

Integrate the density to accumulate probability, differentiate the CDF to get the density back.

Which one to reach for

Use the PDF when you need the shape: where the mass concentrates, whether it is skewed, how fat the tails are.

Use the CDF when you need a probability, and in practice that is most of the time:

P(a≤X≤b)=F(b)−F(a)P(X>a)=1−F(a)P(a \leq X \leq b) = F(b) - F(a) \qquad P(X > a) = 1 - F(a)

Subtracting two CDF values beats evaluating an integral, which is why tables and library functions give you the CDF.

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