Difference between revisions of "Notes:Richard Sharp question"

Revision as of 20:07, 25 November 2016

Problem

I wish to show:

• [math]2\Vert f\Vert_\infty \sum_{k:(k-nx)^2\ge n^2\delta^2}{}^nC_kx^k(1-x)^{n-k}\le 2\Vert f\Vert_\infty \frac{1}{n^2\delta^2}\sum_{k\eq 0}^n(k-nx)^2\ {}^nC_kx^k(1-x)^{n-k}[/math]

I am completely happy with the LHS, I understand the RHS is probably best written as:

• [math]2\Vert f\Vert_\infty \sum_{k\eq 0}^n\frac{(k-nx)^2}{n^2\delta^2}\ {}^nC_kx^k(1-x)^{n-k} [/math]

By hypothesis, for the summation on the LHS, [ilmath]k[/ilmath] is such that:

• [ilmath](k-nx)^2\ge n^2\delta^2\implies\frac{(k-nx)^2}{n^2\delta^2}\ge 1[/ilmath]

I cannot see how to get to the RHS

Supporting equations

Note that:

1. [math]\sum^n_{k\eq 0}{}^nC_kx^k(1-x)^{n-x}\eq 1[/math]
2. [math]\sum^n_{k\eq 0}(k-nx)^2\ {}^nC_kx^k(1-x)^{n-1} \eq nx(1-x)[/math]
3. [math]\sum^n_{k\eq 0}k\ {}^nC_kx^k(1-x)^{n-1}\eq nx[/math] - probably not important