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Some Continuous Probability Distributions

This review from Math 727 covers various topics such as almost sure convergence, convergence in probability, quadratic mean convergence, uniform integrability, independent and identically distributed random variables, Chebyshev inequality, and Gamma, Chi-Squared, Beta distributions.

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Some Continuous Probability Distributions

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  1. Some Continuous Probability Distributions Asmaa Yaseen

  2. Review from Math 727 • Convergence of Random Variables The almost sure convergence The sequence converges to almost surely denoted by , if

  3. Review from Math 727 • The convergence in Probability The sequence converges to X in probability denoted by , if

  4. Review from Math 727 Quadratic Mean Convergence Almost Sure Convergence Convergence in Uniform integrability Convergence in probability Constant limit Convergence in distribution

  5. Review from Math 727 • Let be a sequence of independent and identically distributed random variables, each having a meanandstandard deviation . Define a new variable Then, as , the sample mean X  equals the populationmean of each variable

  6. Review from Math 727

  7. Review from Math 727 In addition

  8. Review from Math 727 • Therefore, by the Chebyshev inequality, for all , As , it then follows that

  9. Gamma, Chi-Squared ,Beta Distribution Gamma Distribution The Gamma Function for The continuous random variable X has a gamma distribution, with parameters α and β, if its density function is given by 0, Otherwise

  10. Gamma, Chi-Squared ,Beta Distribution Gamma’s Probability density function

  11. Gamma, Chi-Squared ,Beta Distribution Gamma Cumulative distribution function

  12. Gamma, Chi-Squared ,Beta Distribution The mean and variance of the gamma distribution are :

  13. Gamma, Chi-Squared ,Beta Distribution The Chi- Squared Distribution The continuous random variable X has a chi-squared distribution with v degree of freedom, if its density function is given by Elsewhere ,

  14. Gamma, Chi-Squared ,Beta Distribution

  15. Gamma, Chi-Squared ,Beta Distribution

  16. Gamma, Chi-Squared ,Beta Distribution • The mean and variance of the chi-squared distribution are Beta Distribution It an extension to the uniform distribution and the continuous random variable X has a beta distribution with parameters and

  17. Gamma, Chi-Squared ,Beta Distribution If its density function is given by The mean and variance of a beta distribution with parameters α and β are

  18. Gamma, Chi-Squared ,Beta Distribution

  19. Gamma, Chi-Squared ,Beta Distribution

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