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Understanding Mean, Variance, and Standard Deviation in Discrete Random Variables

This guide covers the concepts of mean (expected value), variance, and standard deviation for discrete random variables. It explains how to calculate the mean using the formula µ = ΣxP(x), where each value is multiplied by its probability, and discusses the expected value using a lottery example with 1 ticket sold from 1500. Additionally, it provides formulas for variance (σ² = Σ(x - µ)²P(x)) and standard deviation (σ = √σ²). Learn through multiple examples how to apply these concepts effectively.

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Understanding Mean, Variance, and Standard Deviation in Discrete Random Variables

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  1. 4.1B – Probability Distribution • MEAN of discrete random variable: µ = ΣxP(x) • EACH x is multiplied by its probability and the products are added. • µ = EXPECTED VALUE of discrete random variables

  2. Example: Find the Mean

  3. Example: Find the Mean

  4. Example: Find the Mean

  5. Example: Find the Mean

  6. Example: Find the Mean

  7. Example: Find the Mean

  8. Example: Find the Mean

  9. Example: Find the Mean

  10. Example: Find the Mean

  11. Example: Find the Mean

  12. Example: Find the Mean

  13. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  14. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  15. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  16. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  17. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  18. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  19. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  20. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  21. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  22. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  23. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  24. Example: Find the EXPECTED VALUE of your gain. • 1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?

  25. Standard Deviation • VARIANCE of discrete random variable σ² = Σ(x-µ)²P(x) OR σ² = [Σx²P(x)] - µ² • STANDARD DEVIATION of discrete random variable σ = √σ²

  26. Example: Find Variance & Standard Deviation σ = √σ²

  27. Example: Find Variance & Standard Deviation σ = √σ²

  28. Example: Find Variance & Standard Deviation σ = √σ²

  29. Example: Find Variance & Standard Deviation σ = √σ²

  30. Example: Find Variance & Standard Deviation σ = √σ²

  31. Example: Find Variance & Standard Deviation σ = √σ²

  32. Example: Find Variance & Standard Deviation σ = √σ²

  33. Example: Find Variance & Standard Deviation σ = √σ²

  34. Example: Find Variance & Standard Deviation σ = √σ²

  35. Example: Find Variance & Standard Deviation σ = √σ²

  36. Example: Find Variance & Standard Deviation σ = √σ²

  37. Example: Find Variance & Standard Deviation σ = √σ²

  38. Example: Find Variance & Standard Deviation σ = √σ²

  39. Example: Find Variance & Standard Deviation σ = √σ²

  40. Example: Find Variance & Standard Deviation σ = √σ²

  41. Example: Find Variance & Standard Deviation σ = √σ²

  42. Example: Find Variance & Standard Deviation σ = √σ²

  43. Example: Find Variance & Standard Deviation σ = √σ² = √1.616 = 1.27

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