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∙ if we are given one of the conditional densities

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Unformatted text preview: ∙ If we are given one of the conditional densities, say f Y | X y | x , and then the marginal density of X , we can always recover the joint density: f X , Y x , y f Y | X y | x f X x 7 ∙ Once we have a conditional density we can define a conditional CDF. 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 ∙ 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 EXAMPLE : For the population of working-age adults who were unemployed last year, let Y be a Bernoulli random variable indicating whether the person has a job this year. Let X be the number of hours spent in a job training program offered during unemployment – where the maximum possible hours is, say, 120 hours. Suppose the conditional probability of being employed is P Y 1| X x 80 x 120 x 10 Then P Y 1| X 2/3 ≈ .67 P Y 1| X 40 .75 P Y 1| X 80 .80 P Y 1| X 120 .5/6 ≈ .83 ∙ The employment probability increases with hours of job training. 11 .65 .7 .75 .8 .85 P(Y = 1|X = x) 40 80 120 x P(Y = 1|X = x) = (80 + x)/(120 + x) 12 ∙ As a shorthand, it will be useful to let D Y | X denote the distribution of Y given X . For us, we will always mean by this that we have a conditional density, f Y | X y | x , that fully describes this distribution. ( cd1 ) Let X , Y , and Z be random vectors. Then f Y , X | Z y , x | z f Y | X , Z y | x , z f X | Z x | z The left hand side is the joint density of Y , X conditional on Z . Notice that when Z is empty we get the definition of a conditional density. 13 The proof is straightforward by repeated application of the definition: f Y , X | Z y , x | z f Y , X , Z y , x , z f Z z f Y | X , Z y | x , z f X , Z x , z f Z z f Y | X , Z y | x , z f X , Z x , z f Z z f Y | X , Z y | x , z f X | Z x | z ∙ This relationship is very important for finding joint densities of two or more “endogenous” variables ( X and Y ) conditional on some “exogenous” variables ( Z ). 14 ( cd2 ) Suppose we have a conditional densities f Y | X , Z y | x , z and f Z | X z | x we want f Y | X y | x . If Z is discrete then f Y | X y | x ∑ j 1 f Y | X , Z y | x , z j f Z | X z j | x whereas if Z is continuous, f Y | X y | x −...
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∙ If we are given one of the conditional densities say f...

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