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Chapter 2. Random Variables

Chapter 2. Random Variables. 2.1 Discrete Random Variables 2.2 Continuous Random Variables 2.3 The Expectation of a Random Variable 2.4 The Variance of a Random Variable 2.5 Jointly Distributed Random Variables 2.6 Combinations and Functions of Random Variables. -3. -2. -1. 1. 0. 2. 3.

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Chapter 2. Random Variables

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  1. Chapter 2. Random Variables 2.1 Discrete Random Variables 2.2 Continuous Random Variables 2.3 The Expectation of a Random Variable 2.4 The Variance of a Random Variable 2.5 Jointly Distributed Random Variables 2.6 Combinations and Functions of Random Variables

  2. -3 -2 -1 1 0 2 3 2.1 Discrete Random Variable2.1.1 Definition of a Random Variable (1/2) • Random variable • A numerical value to each outcome of a particular experiment

  3. 2.1.1 Definition of a Random Variable (2/2) • Example 1 : Machine Breakdowns • Sample space : • Each of these failures may be associated with a repair cost • State space : • Cost is a random variable : 50, 200, and 350

  4. 2.1.2 Probability Mass Function (1/2) • Probability Mass Function (p.m.f.) • A set of probability value assigned to each of the values taken by the discrete random variable • and • Probability :

  5. 0.5 0.3 0.2 50 200 350 2.1.2 Probability Mass Function (1/2) • Example 1 : Machine Breakdowns • P (cost=50)=0.3, P (cost=200)=0.2, P (cost=350)=0.5 • 0.3 + 0.2 + 0.5 =1

  6. 1.0 0.5 0.3 0 50 200 350 2.1.3 Cumulative Distribution Function (1/2) • Cumulative Distribution Function • Function : • Abbreviation : c.d.f

  7. 2.1.3 Cumulative Distribution Function (2/2) • Example 1 : Machine Breakdowns

  8. 2.2 Continuous Random Variables2.2.1 Example of Continuous Random Variables (1/1) • Example 14 : Metal Cylinder Production • Suppose that the random variable is the diameter of a randomly chosen cylinder manufactured by the company. Since this random variable can take any value between 49.5 and 50.5, it is a continuous random variable.

  9. 2.2.2 Probability Density Function (1/4) • Probability Density Function (p.d.f.) • Probabilistic properties of a continuous random variable

  10. 49.5 50.5 2.2.2 Probability Density Function (2/4) • Example 14 • Suppose that the diameter of a metal cylinder has a p.d.f

  11. 2.2.2 Probability Density Function (3/4) • This is a valid p.d.f.

  12. 49.8 50.1 49.5 50.5 2.2.2 Probability Density Function (4/4) • The probability that a metal cylinder has a diameter between 49.8 and 50.1 mm can be calculated to be

  13. 2.2.3 Cumulative Distribution Function (1/3) • Cumulative Distribution Function

  14. 2.2.2 Probability Density Function (2/3) • Example 14

  15. 2.2.2 Probability Density Function (3/3) 1 49.7 50.0 50.5 49.5

  16. 2.3 The Expectation of a Random Variable2.3.1 Expectations of Discrete Random Variables (1/2) • Expectation of a discrete random variable with p.m.f • Expectation of a continuous random variable with p.d.f f(x) • The expected value of a random variable is also called the mean of the random variable

  17. 2.3.1 Expectations of Discrete Random Variables (2/2) • Example 1 (discrete random variable) • The expected repair cost is

  18. 2.3.2 Expectations of Continuous Random Variables (1/2) • Example 14 (continuous random variable) • The expected diameter of a metal cylinder is • Change of variable: y=x-50

  19. 2.3.2 Expectations of Continuous Random Variables (2/2) • Symmetric Random Variables • If has a p.d.f that is symmetric about a point so that • Then, (why?) • So that the expectation of the random variable is equal to the point of symmetry

  20. 2.3.3 Medians of Random Variables (1/2) • Median • Information about the “middle” value of the random variable • Symmetric Random Variable • If a continuous random variable is symmetric about a point , then both the median and the expectation of the random variable are equal to

  21. 2.3.3 Medians of Random Variables (2/2) • Example 14

  22. 2.4 The variance of a Random Variable2.4.1 Definition and Interpretation of Variance (1/2) • Variance( ) • A positive quantity that measures the spread of the distribution of the random variable about its mean value • Larger values of the variance indicate that the distribution is more spread out • Definition: • Standard Deviation • The positive square root of the variance • Denoted by

  23. 2.4.1 Definition and Interpretation of Variance (2/2) Two distribution with identical mean values but different variances

  24. 2.4.2 Examples of Variance Calculations (1/1) • Example 1

  25. 2.4.3 Chebyshev’s Inequality (1/1) • Chebyshev’s Inequality • If a random variable has a mean and a variance , then • For example, taking gives

  26. Proof

  27. 2.4.4 Quantiles of Random Variables (1/2) • Quantiles of Random variables • The th quantile of a random variable X • A probability of that the random variable takes a value less than the th quantile • Upper quartile • The 75th percentile of the distribution • Lower quartile • The 25th percentile of the distribution • Interquartile range • The distance between the two quartiles

  28. 2.4.4 Quantiles of Random Variables (2/2) • Example 14 • Upper quartile : • Lower quartile : • Interquartile range :

  29. 2.5 Jointly Distributed Random Variables2.5.1 Jointly Distributed Random Variables (1/4) • Joint Probability Distributions • Discrete • Continuous

  30. 2.5.1 Jointly Distributed Random Variables (2/4) • Joint Cumulative Distribution Function • Discrete • Continuous

  31. 2.5.1 Jointly Distributed Random Variables (3/4) • Example 19 : Air Conditioner Maintenance • A company that services air conditioner units in residences and office blocks is interested in how to schedule its technicians in the most efficient manner • The random variable X, taking the values 1,2,3 and 4, is the service time in hours • The random variable Y, taking the values 1,2 and 3, is the number of air conditioner units

  32. 2.5.1 Jointly Distributed Random Variables (4/4) • Joint p.m.f • Joint cumulative distribution function

  33. 2.5.2 Marginal Probability Distributions (1/2) • Marginal probability distribution • Obtained by summing or integrating the joint probability distribution over the values of the other random variable • Discrete • Continuous

  34. 2.5.2 Marginal Probability Distributions (2/2) • Example 19 • Marginal p.m.f of X • Marginal p.m.f of Y

  35. Example 20: (a jointly continuous case) • Joint pdf: • Marginal pdf’s of X and Y:

  36. 2.5.3 Conditional Probability Distributions (1/2) • Conditional probability distributions • The probabilistic properties of the random variable X under the knowledge provided by the value of Y • Discrete • Continuous • The conditional probability distribution is a probability distribution.

  37. 2.5.3 Conditional Probability Distributions (2/2) • Example 19 • Marginal probability distribution of Y • Conditional distribution of X

  38. 2.5.4 Independence and Covariance (1/5) • Two random variables X and Y are said to be independent if • Discrete • Continuous • How is this independency different from the independence among events?

  39. 2.5.4 Independence and Covariance (2/5) • Covariance • May take any positive or negative numbers. • Independent random variables have a covariance of zero • What if the covariance is zero?

  40. 2.5.4 Independence and Covariance (3/5) • Example 19 (Air conditioner maintenance)

  41. 2.5.4 Independence and Covariance (4/5) • Correlation: • Values between -1 and 1, and independent random variables have a correlation of zero

  42. 2.5.4 Independence and Covariance (5/5) • Example 19: (Air conditioner maintenance)

  43. What if random variable X and Y have linear relationship, that is, where That is, Corr(X,Y)=1 if a>0; -1 if a<0.

  44. 2.6 Combinations and Functions of Random Variables2.6.1 Linear Functions of Random Variables (1/4) • Linear Functions of a Random Variable • If X is a random variable and for some numbers thenand • Standardization -If a random variable has an expectation of and a variance of , has an expectation of zero and a variance of one.

  45. 2.6.1 Linear Functions of Random Variables (2/4) • Example 21:Test Score Standardization • Suppose that the raw score X from a particular testing procedure are distributed between -5 and 20 with an expected value of 10 and a variance 7. In order to standardize the scores so that they lie between 0 and 100, the linear transformation is applied to the scores.

  46. 2.6.1 Linear Functions of Random Variables (3/4) • For example, x=12 corresponds to a standardized score of y=(4ⅹ12)+20=68

  47. 2.6.1 Linear Functions of Random Variables (4/4) • Sums of Random Variables • If and are two random variables, thenand • If and are independent, so that then

  48. Properties of

  49. 2.6.2 Linear Combinations of Random Variables (1/5) • Linear Combinations of Random Variables • If is a sequence of random variables and and are constants, then • If, in addition, the random variables are independent, then

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