Expectation

Introduction to Probability

4a. Expectation

4. Expectation and variance

Expectation: equivalent definitions.
- [Si] "Expected value" : Sect 1 "Definition and properties" : Item "Definitions"; also Item "A general definition of expected value" (find it before "Jensen's inequality").
- [BT] Sect 2.4 "Expectation, mean, and variance" (pp. 81-83).
- [GS] : Sect 6.1 "Expected value of discrete random variables" (pp. 229-230).
Some interpretations and paradoxes.
- [Si] "Expected value" : Sect 1 "Definition and properties" : Item "Interpretation".
- [Si] "Bernoulli trials" : Sect 3 "The geometric distribution" : Items "The Peterburg problem", "The alternating coin-tossing game".
- [BT] Sect 2.4 "Expectation, mean, and variance" (pp. 81-83).
- [GS] : Sect 6.1 "Expected value of discrete random variables" (pp. 227-229, 249-250).
Expectation of an indicator.
- [Si] "Expected value" : Sect 1 "Definition and properties" : Item "Examples and applications" (find it after Exercise 11).
Expectation and symmetry.
Expectation of uniform random variables.
- [Si] "Expected value" : Sect 1 "Definition and properties" : Item "Examples and applications" (find it after Exercise 11).
- [BT] Sect 2.4 "Expectation, mean, and variance" : Example 2.6 (p. 88); also Sect. 2.8 "Summary and discussion".
Expectation of a function: change of variable.
- [Si] "Expected value" : Sect 1 "Definition and properties" : Item "Change of variable theorem".
- [BT] Sect 2.4 "Expectation, mean, and variance" : Item "Expected value rule for functions of random variables (p. 84).
- [GS] : Sect 6.1 "Expected value of discrete random variables" (pp. 229-230).
Linearity of expectation.
- [Si] "Expected value" : Sect 1 "Definition and properties" : Item "linearity" (find it after "Change of variable theorem").
- [BT] Sect 2.5 "Joint PMF's of multiple random variables" : Item "Functions of multiple random variables" (p. 94-).
- [GS] : Sect 6.1 "Expected value of discrete random variables" (pp. 230-231).
Sums of indicators.
- [GS] : Sect 6.1 "Expected value of discrete random variables" (pp. 232, 234-235).
Expectation of binomial, Poisson, and hypergeometric random variables.
- [Si] "Bernoulli trials" : Sect 2 "The binomial distribution" : Item "Moments".
- [Si] "Finite sampling models" : Sect 2 "The hypergeometric distribution" : Item "Moments".
- [BT] Sect 2.4 "Expectation, mean, and variance" : Item "mean and variance of some common random variables" (p. 88-); also Sect 2.5 "Joint PMF's of multiple random variable" : Example 2.10, and Sect. 2.8 "Summary and discussion" (p. 116-).
- [GS] : Sect 6.1 "Expected value of discrete random variables" (p. 233, binomial, Poisson; p. 250, Poisson).
Markov inequality.
- [Si] "Expected value" : Sect 1 "Definition and properties" : Item "Special results for nonnegative variables".
- [BT] Sect 7.1 "Markov and Chebyshev inequalities" (p. 381).
Jensens's inequality.
- [Si] "Expected value" : Sect 1 "Definition and properties" : Item "Jensen's inequality".

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