# Marginal Distributions

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## Marginal Distributions

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IV. Marginal Distributions V. Expectations and Variance VI. Covariance

- Lecture number:
- 2
- Pages:
- 2
- Type:
- Lecture Note
- School:
- Cornell University
- Course:
- Econ 3120 - Applied Econometrics
- Edition:
- 1

**Unformatted text preview: **

Econ 3120 1st Edition Lecture 2 Outline of Last Lecture I Random Variables II Continuous Distributions III Multivariate Distributions Outline of Current Lecture IV Marginal Distributions V Expectations and Variance VI Covariance Current Lecture 1 4 Marginal Distributions Suppose we have a joint distribution of X and Y given by f x y How do we find the distribution of Y alone The marginal distribution of a discrete random variable X is given by g x y f x y Similarly the marginal distribution of Y is given by h y x f x y In the continuous case the marginal distributions for X and Y are given by g x f x y dy and h y f x y dx 1 5 Conditional Distributions The conditional distribution of X is defined as f x y f x y h y when h y is the value of the marginal distribution of Y at y The conditional distribution of Y is defined similarly Given the marginal distribution for X of g x the conditional distribution is given by w y x f x y g x 1 6 Independence Random variables X and Y are independent if and only if f x y f x f y f x y g x Expectation and Variance 2 1 Expectation Definition If X is a discrete random variable and f x represents its probability distrubtion function the expected value or mean of X is given by E X x x x f x For a continuous random variable the expected value is given by E X x x f x dx We can think about the expected value as the weighted average of X The value of each possible realization of X is weighted by the probability that x occurs We can also take expectations of functions of a random variable E w X x w x f x dx Or we can take expectations of functions of multiple random variables E w X Y x w x y f x y dxdy 2 1 1 Properties of Expectations 1 For any constant c E c c 2 For any constants a and b E aX b aE X b 3 If a1 a2 an are constants and X1 X2 Xn are random variables then E n i 1 aiXi aiE Xi These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute 2 2

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