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Marginal and Conditional Distributions of Continuous Random Variables

This lecture continues Chapter 5 on statistics and covers the definitions of marginal and conditional probability density functions for continuous random variables. It includes examples on finding marginal and conditional distributions, independence of random variables, and conditional probability.

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Marginal and Conditional Distributions of Continuous Random Variables

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  1. Statistics 270 - Lecture 19

  2. Continue Chapter 5 today

  3. Definition • The marginal probability density function for continuous random variables X and Y, denoted by fX(x) and fY(y), respectively, are given by

  4. Example: • The front tire on a particular type of car is suppose to be filled to a pressure of 26 psi • Suppose the actual air pressure in EACH tire is a random variable (X for the right side; Y for the left side) with joint pdf • Find the marginal distribution of X

  5. More Generally

  6. Independence • Two random variables X and Y are said to be independent if: • Discrete: • Continuous:

  7. Example • Consider 3 continuous random variables X,Y, and Z with joint pdf: • X, Y and Z independent?

  8. Conditional Probability • Let X and Y be rv’s with joint pdf f(x,y) and marginal distribution of X, f(x). The the conditional probability density of Y, given X=x is defined as • Substitute the pmf’s if X and Y are discrete

  9. Example • Consider 2 continuous random variables X, and Y with joint pdf: • Find the marginal distribution of X • Find the conditional distribution of Y given X=0.2

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