418531 ภาคต้น 2552/โจทย์ปัญหาความน่าจะเป็น II/เฉลยข้อ 7

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Error

Too many requests (f061ab2)

If you report this error to the Wikimedia System Administrators, please include the details below.

Request served via cp5021 cp5021, Varnish XID 113677314
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Wed, 17 Jun 2026 04:10:57 GMT


Sensitive client information

IP address: 158.108.32.49

$X_{i}=|a_{i}-i|\,$ จาก law of total expectation เราได้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 109386177 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 17 Jun 2026 04:10:52 GMT Sensitive client information IP address: 158.108.32.49 $E[X_{i}]\,$	$=\,$	$\Pr(a_{i}<i)E[X_{i}\ |\ a_{i}<i]+\Pr(a_{i}=i)E[X_{i}\ |\ a_{i}=i]+\Pr(a_{i}>i)E[X_{i}\ |\ a_{i}>i]\,$
	$=\,$
	$=\,$
		${\frac {i-1}{n}}{\bigg (}i-{\frac {i}{2}}{\bigg )}+{\frac {n-i}{n}}{\bigg (}{\frac {n+i+1}{2}}-i{\bigg )}\,$
	$=\,$	${\frac {i-1}{n}}\cdot {\frac {i}{2}}+{\frac {n-i}{n}}\cdot {\frac {n-i+1}{2}}\,$
	$=\,$

ฉะนั้น

	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 114528757 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 17 Jun 2026 04:10:54 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{i=1}^{n}E[X_{i}]\,$
	$=\,$	${\frac {1}{n}}{\bigg (}\sum _{i=1}^{n}i^{2}-\sum _{i=1}^{n}i{\bigg )}\,$
		${\frac {1}{n}}{\bigg (}{\frac {n(n+1)(2n+1)}{6}}-{\frac {n(n+1)}{2}}{\bigg )}\,$
	$=\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 97204875 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 17 Jun 2026 04:10:55 GMT Sensitive client information IP address: 158.108.32.49 $(n+1){\frac {2n-2}{6}}\,$
	$=\,$	${\frac {n^{2}-1}{3}}\,$