Or we can start with a function that satisfies the

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Or, we can start with a function that satisfies the properties of a CDF for each x and then obtain a PDF – in the usual ways for discrete, continuous, or mixed random variables. 8
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One sometimes starts with a joint distribution, finds the marginal of X , and then obtains f Y | X y | x using the formula. But in the social sciences it is much more common to directly specify f Y | X y | x so that f Y | X .| x is a PDF for all x , often being silent on the distribution of X . The idea is we are interested in how f Y | X .| x changes with x – for example, how does the earnings distribution change with different levels of education? 9
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