Unit 4-Lecture 5: Expectation

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本文介绍了概率论中的核心概念——期望，并详细探讨了几种常见的随机变量分布，包括几何分布和二项分布等。此外，还讲解了条件期望、总期望定律及随机变量线性组合的期望计算。

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If R is a random variable defined on a sample space S, then the expectation of R is:
$E x [R] : : = \sum w \in S R (ω) P r [ω]$ $Ex[R] ::= \sum_{w\in S}R(\omega)Pr[\omega]$
Alternate Definition for any random variable R:
$E x [R] = \sum x \in r a n g e (R) x \cdot P r [R = x]$ $Ex[R] = \sum_{x \in range(R)}x\cdot Pr[R=x]$

The conditional expectation Ex[R|A] of a random variable R given event A is:
$E x [R | A] : : = \sum r \in r a n g e (R) r \cdot P r [R = r | A]$ $Ex[R|A] ::= \sum_{r\in range(R)}r \cdot Pr[R=r|A]$

Let R be a random variable on a sample space S, and suppose that A1, A2, … , is a partition of S. Then
$E x [R] = \sum i E x [R | A i] \cdot P r [A i]$ $Ex[R] = \sum_iEx[R|A_i]\cdot Pr[A_i]$

If a system independently fails at each time step with probability p, then the expected number of steps up to the first failure is $\frac{1}{p}$ .

A random variable, C , has a geometric distribution with parameter p iff $codomain(C) = \mathbb Z^+$ and:
$P r [C = i] = (1 - p) i - 1 p$ $Pr[C=i] =(1-p)^{i-1}p$
If a random variable C has a geometric distribution with parameter p, then:
$E x [C] = 1 p$ $Ex[C] = \frac{1}{p}$

For any random variables R1 and R2,
$E x [a 1 R 1 + a 2 R 2] = a 1 E x [R 1] + a 2 E x [R 2]$ $Ex[a_1R_1+a_2R_2] = a_1Ex[R_1] + a_2Ex[R_2]$
Sums of Indicator Random Variables:
Linearity of expectation is especially useful when you have a sum of indicator random variables

If random variables $R_1, R_2,...,R_k$ are mutually independent, then
$E (\prod i = 1 k R i) = \prod i = 1 k E x [R i]$ $E(\prod_{i=1}^{k}R_i) = \prod_{i=1}^{k}E_x[R_i]$