βž•Math

5. Joint distribution

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Jun 21, 2023
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mat2017-5
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Statistics
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βž•Math
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Jun 21, 2023 03:18 AM
5. Joint distributionJoint distribution: DiscreteJoint PMFConditional PMFExampleLOTUSIndependence of discrete r.v.sSum of Poissons(Chicken-egg problem)ExampleJoint distribution: ContinuousJoint CDF, PDFMarginal, Conditional PDFIndependence of discrete r.v.sTransformed joint r.v.sExampleExpected; unifExpected; normExpected; expoLightbulbTwo busesCauchy PDFExampleCovarianceCovarianceCorrelationMax(X,Y), Min(X,Y)Scientist

5. Joint distribution

Joint distribution: Discrete

Joint PMF

Conditional PMF

Example

  • triangle in the plane: (0, 0), (1, 0), (0, 1)
  • area: , joint PDF(probability):
  • marginal PDF of X:
  • conditional PDF of X given Y:

LOTUS

Β 
Β 

Independence of discrete r.v.s

Sum of Poissons(Chicken-egg problem)

  • , , and X and Y are independent, then
  • , , , then X and Y are independent

Example

  • , ,

Joint distribution: Continuous

Joint CDF, PDF

Marginal, Conditional PDF

Independence of discrete r.v.s

Transformed joint r.v.s

Example

Expected; unif

Expected; norm

Expected; expo

  • with memoryless property

Lightbulb

A lightbulb was man- ufactured by one of two companies. Bulbs that are made by Company 0 last an Expo(Ξ»0) amount of time, and bulbs made by Company 1 last an Expo(Ξ»1) amount of time, with Ξ»0 < Ξ»1. The bulb of interest here was made by Company 0 with probability p0 and by Company 1 with probability p1 = 1 βˆ’ p0, but from inspecting the bulb we don’t know which company made it.
  • T: how long the bulb lasts

Two buses

  • One bus: , the other one:
  • The other one first arrive:
  • CDF of waiting time:

Cauchy PDF

Example

Covariance

Covariance

Correlation

Max(X,Y), Min(X,Y)

Generalization

Scientist

A scientist makes two measurements, considered to be independent standard Normal r.v.s. Find the correlation between the larger and smaller of the values.