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probability - Find expected value using CDF - Cross Validated @styfle - because that's what a PDF is, whenever the CDF is continuous and differentiable You can see this by looking at how you have defined your CDF Differentiating an integral just gives you the integrand when the upper limit is the subject of the differentiation
distributions - Empirical CDF vs CDF - Cross Validated The CDF is a theoretical construct - it is what you would see if you could take infinitely many samples The empirical CDF usually approximates the CDF quite well, especially for large samples (in fact, there are theorems about how quickly it converges to the CDF as the sample size increases)