{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

slides_5_conddist

# ∙ if we are given one of the conditional densities

This preview shows pages 7–16. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 −...
View Full Document

{[ snackBarMessage ]}

### Page7 / 67

∙ If we are given one of the conditional densities say f...

This preview shows document pages 7 - 16. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online