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Chapter 5 Expectations : Slide 2 Content Introduction Expectation of a Function of a Random Variable Expectation of Functions of Multiple Random Variables Important Properties of Expectation Conditional Expectations Moment Generating Functions Inequalities The Weak Law of Large Numbers and Central Limit Theorems Slide 3 Introduction Chapter 5 Expectations Slide 4 Slide 5 Slide 6 Definition Expectation The expectation (mean), E[X] or X, of a random variable X is defined by: Slide 7 Definition Expectation The expectation (mean), E[X] or X, of a random variable X is defined by: provided that the relevant sum or integral is absolutely convergent, i.e., Slide 8 Definition Expectation The expectation (mean), E[X] or X, of a random variable X is defined by: provided that the relevant sum or integral is absolutely convergent, i.e., Slide 9 Example 1 Let X denote #good components in the experiment. Slide 10 Example 2 Slide 11 Example 3 pdf Slide 12 Example 3 Slide 13 Expectation of a Function of a Random Variable Chapter 5 Expectations Slide 14 The Expectation of Y=g(X) Slide 15 Slide 16 Example 4 Slide 17 Example 5 Slide 18 Moments g(X) k k (mean) (variance) Slide 19 X : Slide 20 Example 6 X ~ B(n, p) E[X]=? Var[X]=? Slide 21 Example 6 X ~ B(n, p) E[X]=? Var[X]=? Slide 22 Example 6 X ~ B(n, p) E[X]=? Var[X]=? Slide 23 Example 7 X ~ Exp( ) E[X]=? Var[X]=? Slide 24 Summary of Important Moments of Random Variables Slide 25 Expectation of Functions of Multiple Random Variables Chapter 5 Expectations Slide 26 The Expectation of Y = g(X 1, , X n ) Slide 27 Example 8 X Y p(x, y) Slide 28 Example 9 Slide 29 Important Properties of Expectation Chapter 5 Expectations Slide 30 Linearity E1. E2. X 1, X 2, , X n Slide 31 Example 10 X Y E[X+Y] = E[X]+E[Y]. Slide 32 A Question X Y E[X+Y] = E[X]+E[Y]. ? Slide 33 Independence E3. If random variables X 1,..., X n are independent, then Slide 34 Example 11 X Y E[XY] = E[X]E[Y]. Slide 35 A Question X Y E[XY] = E[X]E[Y]. X Y ? Slide 36 Example 12 X Y Slide 37 A Question ? Slide 38 The Variance of Sum Define Slide 39 The Variance of Sum Slide 40 The Covariance Slide 41 The Covariance Slide 42 Example 13 Slide 43 A Question X Y ? Slide 44 Properties Related to Covariance E4. E5. Slide 45 Properties Related to Covariance E4. E5. Fact: Slide 46 Properties Related to Covariance E4. E5. E6. E7. Slide 47 Example 14 Slide 48 Slide 49 More Properties on Covariance E8. Slide 50 More Properties on Covariance E8. E9. Slide 51 Example 16 Slide 52 Slide 53 Slide 54 Theorem 1 Schwartz Inequality Slide 55 Pf) E = * E Slide 56 Theorem 1 Schwartz Inequality Pf) E Slide 57 Theorem 1 Schwartz Inequality Pf) E Slide 58 Corollary E10. Pf) Slide 59 Correlation Coefficient E11. Slide 60 Correlation Coefficient E11. Fact: Is the converse also true? Slide 61 Correlation Coefficient E11. E12. Pf) 0 0 Slide 62 Example 18 Slide 63 Slide 64 Slide 65 Example 19 2 X: # Y: # Slide 66 Example 19 2 X: # Y: # Method 1: X Y p(x, y) Slide 67 Example 19 2 X: # Y: # Method 2: Facts: Slide 68 Conditional Expectations Chapter 5 Expectations Slide 69 Definition Conditional Expectations Slide 70 Facts a function of X (x) See text for the proof E13. Slide 71 Conditional Variances Slide 72 Example 20 Slide 73 Moment Generating Functions Chapter 5 Expectations Slide 74 Moment Generating Functions Moments Moments Slide 75 Moment Generating Functions The moment generating function M X (t) of a random variable X is defined by The domain of M X (t) is all real numbers such that e Xt has finite expectation. Slide 76 Example 21 Slide 77 Example 22 Slide 78 Summary of Important Moments of Random Variables Slide 79 Moment Generating Functions The moment generating function M X (t) of a random variable X is defined by The domain of M X (t) is all real numbers such that e Xt has finite expectation. M X (t) ? Slide 80 Moment Generating Functions Slide 81 0 0 1 1 2 2 k k Slide 82 0 0 1 1 2 2 k k Slide 83 Slide 84 Example 23 Using MGF to find the means and variances of Slide 85 Example 23 Slide 86 Slide 87 Slide 88 Slide 89 Correspondence or Uniqueness Theorem Let X 1, X 2 be two random variables. Slide 90 Example 24 Slide 91 Slide 92 Slide 93 Slide 94 Slide 95 Theorem Linear Translation Pf) Slide 96 Theorem Convolution Pf)... Slide 97 Example 25... Slide 98 Example 25... Slide 99 Example 25... Slide 100 Example 25... Slide 101 Example 25... Slide 102 Example 26 Slide 103 0 Slide 104 0 Slide 105 Theorem of Random Variables Sum Slide 106 We have proved the above five using probability generating functions. They can also be proved using moment generating functions. Slide 107 Theorem of Random Variables Sum Slide 108 Slide 109 Slide 110 Slide 111 Slide 112 Slide 113 Slide 114 Slide 115 Inequalities Chapter 5 Expectations Slide 116 Theorem Markov Inequality Let X be a nonnegative random variable with E[X] = . Then, for any t > 0, Slide 117 Theorem Markov Inequality Define A discrete random variable Why? Slide 118 Theorem Markov Inequality Define A discrete random variable Slide 119 Example 27 MTTF Mean Time To Failure Slide 120 Example 27 MTTF Mean Time To Failure By MarkovBy Exponential Distribution Slide 121 Theorem Chebyshev's Inequality Slide 122 Theorem Chebyshev's Inequality Slide 123 Facts: Slide 124 Theorem Chebyshev's Inequality Facts: Slide 125 Example 28 Slide 126 Slide 127 The Weak Law of Large Numbers and Central Limit Theorems Chapter 5 Expectations Slide 128 The Parameters of a Population A population We may never have the chance to know the values of parameters in a population exactly. Slide 129 Sample Mean A population iid random variables iid: identical independent distributions Sample Mean Slide 130 Expectation & Variance of A population Slide 131 Expectation & Variance of A population Slide 132 Expectation & Variance of A population n ? Slide 133 Theorem Weak Law of Large Numbers Let X 1, , X n be iid random variables having finite mean . Slide 134 Theorem Weak Law of Large Numbers Let X 1, , X n be iid random variables having finite mean . Chebyshev's Inequality Slide 135 Central Limit Theorem Let X 1, , X n be iid random variables having finite mean and finite nonzero variance 2. Slide 136 Central Limit Theorem Let X 1, , X n be iid random variables having finite mean and finite nonzero variance 2. Slide 137 Central Limit Theorem Slide 138 Slide 139 = 0 as n Slide 140 Central Limit Theorem n 0 Slide 141 Central Limit Theorem n Slide 142 Central Limit Theorem Slide 143 Slide 144 Let X 1, , X n be iid random variables having finite mean and finite nonzero variance 2. Slide 145 Normal Approximation By the central limit theorem, when a sample size is sufficiently large ( n > 30 ), we can use normal distribution to approximate certain probabilities regarding to the sample or the parameters of its corresponding population. Slide 146 Example 29 Let X i represent the lifetime of i th bulb We want to find n > 30 Slide 147 Example 30 n > 30 Slide 148 Example 30 20 20.5