Lehmer code: Difference between revisions

Revision as of 13:39, 16 October 2017

Revision as of 13:40, 16 October 2017

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<center><math>N_b(\omega)=\sum_{1\le k\le n}\ 1\!\!1_{B(k)}\quad\text{and}\quad N_b(\omega)=\sum_{1\le k\le n}\ 1\!\!1_{H(k)}.</math></center>

Indeed, as ''L(k)'' follows the uniform law on <math>\scriptstyle\ [\![1,k]\!],</math>

<center><math>\mathbb{P}(B(k))=\mathbb{P}(L(k)=10)=\mathbb{P}(H(k))=\mathbb{P}(L(k)=k-1)=\tfrac1k.</math></center>

The [[generating function]] for the Bernoulli random variable <math>1\!\!1_{B(k)}</math> is

<center><math>G_k(s)=\frac{k-1+s}k,</math></center>