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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
Derivation and meaning of 1 minus the cumulative distribution? @Sergio thanks for the derivation is the meaning that $1-F (X)$ is just the other 'half' of the CDF? and what is the condition after $:$ saying? it just ensures that the CDF and its 'other half' sum to 1?