Probability Calculus Anna Janicka - Uniwersytet...
Transcript of Probability Calculus Anna Janicka - Uniwersytet...
![Page 1: Probability Calculus Anna Janicka - Uniwersytet Warszawskicoin.wne.uw.edu.pl/azylicz/pc/pc09_2019_presentation.pdf · 2019-12-02 · Anna Janicka lecture IX, 3.12.2019 RANDOM VECTORS](https://reader033.fdocuments.net/reader033/viewer/2022053017/5f1c3926d053ea4828597cde/html5/thumbnails/1.jpg)
Probability Calculus
Anna Janicka
lecture IX, 3.12.2019
RANDOM VECTORS – cont.
INDEPENDENCE OF RANDOM VARIABLES
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Plan for Today
1. Marginal distributions – cont.
2. Expected values of functions of random
vectors
3. Covariance, correlation
4. Expected value, variance of random
vectors
5. Independence of random variables
properties and characteristics of
independent RV
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Marginal distributions – (cont.)
Marginal distributions of continuous RV:
If marginals are continuous, then the joint
distribution need not be
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Characteristics of random vectors
1. Expected values of functions of the
components of a RV:
2. Examples
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The covariance and correlation coefficient
3. Definitions
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Covariance and correlation coefficient – cont.
4. Properties:
invariance to shifts
bilinearity of the covariance
variance as a special case
simplifying formula:
capture the linear relationship, in other
cases may be misleading
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Correlation coefficient – properties
5. Schwarz inequality
6. Consequences for the correlation coef.
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Expected value and covariance matrix
7. Definitions:
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Properties of EX and the covariance matrix
8. Linearity
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Independent RV
1. Definition of independence
2. Independence of discrete RV
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Independent RV – cont.
3. Example
4. Independence of continuous RV
5. Examples
uniform distribution on square
uniform distribution on circle
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Independent RV – cont. (2)
6. Transformations of RV
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Properties of independent RV
2. Expected value of product
3. Example
4. Covariance of independent RV
5. Non-correlation
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Properties of independent RV – cont.
6. One-way implication only!
independence non-correlation but
IS NOT TRUE!
7. Example – uniform distribution on circle
8. Sum of variances
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Properties of independent RV – cont. (2)
9. Example – sum of points on dice
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