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4-1 Continuous Random Variables

4-1 Continuous Random Variables. 4-2 Probability Distributions and Probability Density Functions. Figure 4-1 Density function of a loading on a long, thin beam. 4-2 Probability Distributions and Probability Density Functions. Figure 4-2 Probability determined from the area under f ( x ).

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4-1 Continuous Random Variables

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  1. 4-1 Continuous Random Variables

  2. 4-2 Probability Distributions and Probability Density Functions Figure 4-1Density function of a loading on a long, thin beam.

  3. 4-2 Probability Distributions and Probability Density Functions Figure 4-2Probability determined from the area under f(x).

  4. 4-2 Probability Distributions and Probability Density Functions Definition

  5. Review pmf

  6. 4-2 Probability Distributions and Probability Density Functions Figure 4-3Histogram approximates a probability density function.

  7. 4-2 Probability Distributions and Probability Density Functions

  8. 4-2 Probability Distributions and Probability Density Functions Example 4-2

  9. 4-2 Probability Distributions and Probability Density Functions Figure 4-5Probability density function for Example 4-2.

  10. 4-2 Probability Distributions and Probability Density Functions Example 4-2 (continued)

  11. 4-3 Cumulative Distribution Functions Definition

  12. 4-3 Cumulative Distribution Functions Example 4-4

  13. Figure 4-7Cumulative distribution function for Example 4-4.

  14. Practice: The pdf for a r.v X is given by f(x) = c x^2, 0≤x≤2 • Find c • Find p(X≤3) • Find CDF of f(x) • Use CDF to find pdf of X

  15. 4-4 Mean and Variance of a Continuous Random Variable Definition

  16. 4-4 Mean and Variance of a Continuous Random Variable Example 4-6

  17. 4-4 Mean and Variance of a Continuous Random Variable Expected Value of a Function of a Continuous Random Variable

  18. 4-4 Mean and Variance of a Continuous Random Variable Example 4-8

  19. 4-5 Continuous Uniform Random Variable Definition

  20. 4-5 Continuous Uniform Random Variable Figure 4-8Continuous uniform probability density function.

  21. 4-5 Continuous Uniform Random Variable Mean and Variance

  22. 4-5 Continuous Uniform Random Variable Example 4-9

  23. 4-5 Continuous Uniform Random Variable Figure 4-9Probability for Example 4-9.

  24. 4-5 Continuous Uniform Random Variable

  25. 4-6 Normal Distribution Definition

  26. 4-6 Normal Distribution Figure 4-10Normal probability density functions for selected values of the parameters  and 2.

  27. 4-6 Normal Distribution Some useful results concerning the normal distribution

  28. 4-6 Normal Distribution Definition : Standard Normal

  29. 4-6 Normal Distribution Example 4-11 Figure 4-13Standard normal probability density function.

  30. 4-6 Normal Distribution

  31. Example:Let Z be a standard normal rv N(0,1), determine the following • p(Z ≤1.26) • p(Z>1.26) • p(Z>-1.26) • P(-1.26<Z<1.26) • Find value z such that p(Z>z) =.05 • Find z such that p(-z<Z<z)=.95

  32. 4-6 Normal Distribution Example 4-13

  33. 4-6 Normal Distribution Figure 4-15Standardizing a normal random variable.

  34. 4-6 Normal Distribution To Calculate Probability

  35. 4-6 Normal Distribution Example 4-14

  36. 4-6 Normal Distribution Example 4-14 (continued)

  37. 4-6 Normal Distribution Example 4-14 (continued) Figure 4-16Determining the value of x to meet a specified probability.

  38. 4-7 Normal Approximation to the Binomial and Poisson Distributions • Under certain conditions, the normal distribution can be used to approximate the binomial distribution and the Poisson distribution.

  39. 4-7 Normal Approximation to the Binomial and Poisson Distributions Figure 4-19Normal approximation to the binomial.

  40. 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-17

  41. 4-7 Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Binomial Distribution

  42. 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-18

  43. 4-7 Normal Approximation to the Binomial and Poisson Distributions Figure 4-21Conditions for approximating hypergeometric and binomial probabilities.

  44. 4-7 Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Poisson Distribution

  45. 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-20

  46. 4-8 Exponential Distribution Definition

  47. 4-8 Exponential Distribution Mean and Variance

  48. 4-8 Exponential Distribution Example 4-21

  49. 4-8 Exponential Distribution Figure 4-23Probability for the exponential distribution in Example 4-21.

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