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Understanding Two-Dimensional Random Variables with Uniform and Multi-Normal Distributions

This article explores the properties and characteristics of two-dimensional random variables, focusing on both uniform and multi-normal distributions. It discusses probability density functions, marginal and conditional probabilities, as well as variance-covariance matrices. The objective is to provide a comprehensive understanding of how to analyze these random variables, including calculating means, standard deviations, and correlations between variables. Practical examples and visualizations enhance the learning experience, making complex concepts more accessible.

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Understanding Two-Dimensional Random Variables with Uniform and Multi-Normal Distributions

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  1. Statistics

  2. A two-dimensional random variable with a uniform distribution

  3. A two-dimensional random variable with a uniform distribution

  4. A two-dimensional random variable with a uniform distribution

  5. A two-dimensional random variable with a uniform distribution

  6. A two-dimensional random variable with a uniform distribution variable 2 variable 1

  7. Probability density function for variable 1 probability value

  8. A two-dimensional random variable with a uniform distribution variable 2 variable 1

  9. Probability density function for variable 2 probability value

  10. A two-dimensional random variable with a multi-normal distribution variable 2 variable 1

  11. A two-dimensional random variable with a multi-normal distribution variable 2 variable 1

  12. A two-dimensional random variable with a multi-normal distribution variable 2 variable 1

  13. A two-dimensional random variable with a multi-normal distribution variable 2 variable 1

  14. A two-dimensional random variable with a multi-normal distribution Marginal probability density function - variable 1 variable 2 variable 1

  15. A two-dimensional random variable with a multi-normal distribution Marginal probability density function - variable 1 variable 2 variable 1 mean

  16. A two-dimensional random variable with a multi-normal distribution Marginal probability density function - variable 1 variable 2 standard deviation variable 1 mean

  17. A single random variable with a multi-normal distribution standard deviation mean

  18. A two-dimensional random variable with a multi-normal distribution variable 2 standard deviation mean marginal probability density function - variable 2 variable 1

  19. A two-dimensional random variable with a multi-normal distribution variable 2 variable 1

  20. A two-dimensional random variable with a multi-normal distribution variable 2 conditional probability density function - variable 1 variable 1

  21. A two-dimensional random variable with a multi-normal distribution variable 2 conditional probability density function - variable 1 variable 1

  22. A two-dimensional random variable with a multi-normal distribution variable 2 conditional probability density function - variable 1 variable 1

  23. Two-dimensional probability density function

  24. To characterise a single random variable we need…..

  25. A single random variable with a multi-normal distribution standard deviation mean

  26. mean value variance σ2 µ

  27. To characterise two random variables we need…..

  28. A two-dimensional random variable with a multi-normal distribution variable 2 variable 1

  29. variance-covariance matrix mean values σ12 σ12 σ21 σ22 µ1 µ2

  30. variance-covariance matrix mean values σ12 σ12 σ21 σ22 µ1 µ2

  31. A two-dimensional random variable with a multi-normal distribution Marginal probability density function - variable 1 variable 2 standard deviation variable 1 mean

  32. A two-dimensional random variable with a multi-normal distribution variable 2 standard deviation mean marginal probability density function - variable 2 variable 1

  33. variance-covariance matrix mean values σ12 σ12 σ21 σ22 µ1 µ2

  34. A two-dimensional random variable with a multi-normal distribution variable 2 Parameter correlation variable 1

  35. A two-dimensional random variable with a multi-normal distribution variable 2 No parameter correlation variable 1

  36. variance-covariance matrix mean values σ12 0 0 σ22 µ1 µ2

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