1. Part V: Continuous Random Variables http://rchsbowman.wordpress.com/2009/11/29 /statistics-notes-%E2%80%93-properties-of-normal-distribution-2/

2. Chapter 23: Probability Density Functions http://divisbyzero.com/2009/12/02 /an-applet-illustrating-a-continuous-nowhere-differentiable-function//

3. Comparison of Discrete vs. Continuous (Examples) DiscreteContinuous Counting: defects, hits, die values, coin heads/tails, people, card arrangements, trials until success, etc. Lifetimes, waiting times, height, weight, length, proportions, areas, volumes, physical quantities, etc.

4. Comparison of mass vs. density Mass (probability mass function, PMF) Density (probability density function, PDF) 0 ≤ p X (x) ≤ 10 ≤ f X (x) P(0 ≤ X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) P(X ≤ 3) ≠ P(X < 3) when P(X = 3) ≠ 0 P(X ≤ 3) = P(X < 3) since P(X = 3) = 0 always

5. Example 1 (class)

6. Example 1 (cont.)

7. Example 2

8. Comparison of CDFs DiscreteContinuous Function graphStep function with jumps of the same size as the mass continuous graphRange: 0 ≤ X ≤ 1

9. Example 3

10. Example 4

11. Example 4 (cont.)

12. Mixed R.V. – CDF Let X denote a number selected at random from the interval (0,4), and let Y = min(X,3). Obtain the CDF of the random variable Y.

13. Chapter 24: Joint Densities http://www.alexfb.com/cgi-bin/twiki/view/PtPhysics/WebHome

14. Probability for two continuous r.v. http://tutorial.math.lamar.edu/Classes/CalcIII/DoubleIntegrals.aspx

15. Example 1 (class) A man invites his fiancée to a fine hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 am and 12 noon. If they arrive a random times during this period, what is the probability that they will meet within 10 minutes? (Hint: do this geometrically)

16. Example: FPF (Cont)

17. Example 2 (class) Consider two electrical components, A and B, with respective lifetimes X and Y. Assume that a joint PDF of X and Y is f X,Y (x,y) = 10e -(2x+5y), x, y > 0 and f X,Y (x,y) = 0 otherwise. a) Verify that this is a legitimate density. b) What is the probability that A lasts less than 2 and B lasts less than 3? c) Determine the joint CDF. d) Determine the probability that both components are functioning at time t. e) Determine the probability that A is the first to fail. f) Determine the probability that B is the first to fail.

18. Example 2d

19. Example 2e

21. Example 3

22. Example 4 (class)

23. Example: Marginal density (class) A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X = the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y = the proportion of time that the walk-up window is in use. The joint PDF is a) What is f X (x)? b) What is f Y (y)?

24. Example: Marginal density (homework) A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is a) What is f X (x)? b) What is f Y (y)?

25. Chapter 25: Independent Why’s everything got to be so intense with me? I’m trying to handle all this unpredictability In all probability -- Long Shot, sung by Kelly Clarkson, from the album All I ever Wanted; song written by Katy Perry, Glen Ballard, Matt Thiessen

26. Example: Independent R.V.’s

27. Example: Independence Consider two electrical components, A and B, with respective lifetimes X and Y with marginal shown densities below which are independent of each other. f X (x) = 2e -2x, x > 0, f Y (y) = 5e -5y, y > 0 and f X (x) = f Y (y) = 0 otherwise. What is f X,Y (x,y)?

28. Example: Independent R.V.’s (homework) A nut company markets cans of deluxe mixed nuts containing almonds, cashews and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. The joint PDF is Are X and Y independent?

29. Chapter 26: Conditional Distributions Q : What is conditional probability? A : maybe, maybe not. http://www.goodreads.com/book/show/4914583-f-in-exams

30. Example: Conditional PDF (class)

31. Chapter 27: Expected values http://www.qualitydigest.com/inside/quality-insider-article /problems-skewness-and-kurtosis-part-one.html#

32. Comparison of Expected Values DiscreteContinuous

33. Example: Expected Value (class) a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time. What is the expected value in each of the following situations:

34. Chapter 28: Functions, Variance http://quantivity.wordpress.com/2011/05/02/empirical-distribution-minimum-variance/

35. Comparison of Functions, Variances DiscreteContinuous Function (general) Function (X 2 ) Variance SD

36. Example: Expected Value - function (class) a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time.

37. Example: Variance (class) a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time. What is the variance in each of the following situations:

38. Friendly Facts about Continuous Random Variables - 1

39. Friendly Facts about Continuous Random Variables - 2

40. Chapter 29: Summary and Review http://www.ux1.eiu.edu/~cfadd/1150/14Thermo/work.html

41. Example: percentile