What is the difference between "probability density function" and "probability distribution function"?

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Whats the difference between probability density function and probability distribution function?

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The relation between the probability density funtion $f$ and the cumulative distribution function $F$ is...

  • if $f$ is discrete: $$ F(k) = \sum_{i \le k} f(i) $$

  • if $f$ is continuous: $$ F(x) = \int_{y \le x} f(y)\,dy $$