418531 ภาคต้น 2552/โจทย์ปัญหาความน่าจะเป็น II/เฉลยข้อ 7

สำหรับ $i=1,2,\ldots ,n\,$ ให้

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 305217699
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:40 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 277869717 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:40 GMT Sensitive client information IP address: 158.108.32.49 $E[X_{i}]\,$
	$=\,$	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 317795388 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:40 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {i-1}{n}}E[i-a_{i}\ |\ a_{i}<i]+{\frac {n-i}{n}}E[a_{i}-i\ |\ a_{i}>i]\,$
	$=\,$
	$=\,$
		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 289627333 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:41 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {i-1}{n}}\cdot {\frac {i}{2}}+{\frac {n-i}{n}}\cdot {\frac {n-i+1}{2}}\,$

ฉะนั้น

$E{\bigg [}\sum _{i=1}^{n}|a_{i}-i|{\bigg ]}\,$
		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 266893664 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:53 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{i=1}^{n}{\frac {1}{2n}}[(i-1)i+(n-i)(n-i+1)]\,$
	$=\,$	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 328014839 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:42 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {1}{n}}\sum _{i=1}^{n}(i-1)i\,$
	$=\,$	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 289169206 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:42 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {1}{n}}{\bigg (}\sum _{i=1}^{n}i^{2}-\sum _{i=1}^{n}i{\bigg )}\,$
	$=\,$	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 314783272 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:43 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {1}{n}}{\bigg (}{\frac {n(n+1)(2n+1)}{6}}-{\frac {n(n+1)}{2}}{\bigg )}\,$
	$=\,$	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 337643397 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:43 GMT Sensitive client information IP address: 158.108.32.49 $(n+1){\bigg (}{\frac {2n+1}{6}}-{\frac {1}{2}}{\bigg )}\,$
	$=\,$	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 312589615 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:44 GMT Sensitive client information IP address: 158.108.32.49 $(n+1){\frac {2n-2}{6}}\,$
	$=\,$	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 277472938 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 07 Sep 2026 09:30:44 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {n^{2}-1}{3}}\,$