如何计算硬币翻转的熵

Question

我想知道......

在重复的硬币翻转中，¿如何计算随机变量X的熵，该熵代表在第一次获得“头部”之前要做的翻转次数？

Answer 1

变量X可以取1到无穷大的任意数字。概率是：

p(X = i) = (1/2)^i

熵是：

H = - Sum {i from 1 to infinity} ( p(X = i) * log2(p(X = i)) )
  = - Sum {i from 1 to infinity} ( 1/2^i * log2(1/2^i) )
  = - Sum {i from 1 to infinity} ( 1/2^i * i * log2(1/2) )
  =   Sum {i from 1 to infinity} ( 1/2^i * i )

解决这个问题：

H = 2 bit