# 线性期望介绍和使用指南

2021年3月18日14:28:51 发表评论 825 次浏览

(1, 1), (1, 2), .... (1, 6)
(2, 1), (2, 2), .... (2, 6)
................
................
(6, 1), (6, 2), ..... (6, 6)

E[R] = r1*p1 + r2*p2 + ... rk*pk

Expected Value of sum = 2*1/36 + 3*1/36 + .... + 7*1/36 +
of two dice throws      3*1/36 + 4*1/36 + .... + 8*1/36 +
........................
.........................
7*1/36 + 8*1/36 + .... + 12*1/36

= 7

E[R1 + R2] = E[R1] + E[R2]

Expected Value of sum of 2 dice throws = 2*(Expected value of one dice throw)
= 2*(1/6 + 2/6 + .... 6/6)
= 2*7/2
= 7

Expected value of sum for n dice throws is = n * 7/2 = 3.5 * n

1. 期望的线性对于相关事件和独立事件均成立。另一方面, 规则E [R1[R2] = E [R1] * E [R2]仅对独立事件有效。
2. 期望的线性对于某个概率空间上的任意数量的随机变量均成立。让R1, R2, R3, ... Rk是k个随机变量, 则
E [R1+ R2+ R3+…+ Rk] = E [R1] + E [R2] + E [R3] +…+ E [Rk]

So the expected number of men to get the right hat back is
= E[R1] + E[R2]  +  .. + E[Rn]
= P(R1 = 1) + P(R2 = 1) + .... + P(Rn = 1)
[Here P(Ri = 1)  indicates probability that Ri is 1]
= 1/n + 1/n + ... + 1/n
= 1

1. 给定一个公平的硬币, 当硬币被抛掷n次时, 预期的正面数是多少。
2. 球和垃圾桶：假设我们有m个球, 标记为i = 1, …, m和n个垃圾桶, 标记为j = 1, .., n。每个球都独立且均匀地随机扔进一个垃圾箱。
a)每个料箱中的预期球数是多少
b)空箱的预期数量是多少。
3. 优惠券收集者：假设彩票中有n种类型的优惠券, 并且每批包含一张优惠券(概率为1 = n)。直到我们每种类型的至少有一张优惠券, 才需要购买多少手(预期)。

http://www.cse.iitd.ac.in/~mohanty/col106/Resources/linearity_expectation.pdf

http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-22-expectation-i/