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Random Variable

Explore the fundamentals of random variables, including discrete and continuous types. Learn about probability mass functions for discrete variables and probability density functions for continuous variables, along with how to calculate expected value and variance. Delve into various probability distributions, including Poisson, Exponential, Uniform, Triangular, and Normal distributions. Understand how these distributions apply to real-world scenarios, such as random arrivals and manufacturing, and the significance of the Central Limit Theorem.

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Random Variable

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  1. Random Variable 2013

  2. Random Variable • Two types • Discrete • Continuous

  3. Random Variable • Probability mass function • Discrete • P(X = xi) = p(xi) •  p(xi) = 1

  4. Random Variable • Probability density function • Continuous • f(x) = e –x x > 0 • P(X = a) = 0 • -  f(x) dx = 1 • P(a < x < b) = ab f(x) dx

  5. Random Variable • Expected value •  = E(x) •  =  xi p (xi) •  =  x f(x) dx

  6. Random Variable • Variance

  7. Random Variable • Standard deviation • Sums of R.V.

  8. Random Variable

  9. Poisson Probability Distribution

  10. Poisson Probability Distribution

  11. Exponential Distribution

  12. Exponential Distribution

  13. The Uniform Distribution

  14. The Uniform Distribution

  15. The Uniform Distribution

  16. The Triangular Distribution • Continuous Distribution

  17. The Triangular Distribution

  18. The Triangular Distribution

  19. Normal Distribution

  20. Normal Distribution

  21. Normal Distribution

  22. Normal Distribution

  23. Selecting a Distribution • Theoretical prior knowledge • Random arrival => exponential IAT • Sum of large manufactures => Normal CLT • Compare histogram with probability mass or probability density

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