418531 ภาคต้น 2552/โจทย์ปัญหาความน่าจะเป็น II/เฉลยข้อ 7

สำหรับ 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 18743355
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sat, 02 May 2026 00:31:43 GMT


Sensitive client information

IP address: 158.108.32.49

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

$E[X_{i}]\,$	$=\,$	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 8916511 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 02 May 2026 00:31:39 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(a_{i}<i)E[X_{i}\ |\ a_{i}<i]+\Pr(a_{i}=i)E[X_{i}\ |\ a_{i}=i]+\Pr(a_{i}>i)E[X_{i}\ |\ a_{i}>i]\,$
	$=\,$	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 1043644202 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 02 May 2026 00:31:39 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(a_{i}<i)E[i-a_{i}\ |\ a_{i}<i]+\Pr(a_{i}=i)\cdot 0+\Pr(a_{i}>i)E[a_{i}-i\ |\ a_{i}>i]\,$
		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 16711807 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 02 May 2026 00:31:45 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {i-1}{n}}E[i-a_{i}\ |\ a_{i}<i]+{\frac {n-i}{n}}E[a_{i}-i\ |\ a_{i}>i]\,$
	$=\,$

ฉะนั้น

$E{\bigg [}\sum _{i=1}^{n}|a_{i}-i|{\bigg ]}\,$	$=\,$
	$=\,$	$\sum _{i=1}^{n}{\frac {1}{2n}}[(i-1)i+(n-i)(n-i+1)]\,$
	$=\,$	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 1054782015 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 02 May 2026 00:31:40 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {1}{2n}}{\bigg (}\sum _{i=1}^{n}(i-1)i+\sum _{i=1}^{n}(n-i)(n-i+1){\bigg )}\,$
	$=\,$	${\frac {1}{2n}}{\bigg (}\sum _{i=1}^{n}(i-1)i+\sum _{i=1}^{n}(i-1)i{\bigg )}\,$
	$=\,$
	$=\,$
		$(n+1){\bigg (}{\frac {2n+1}{6}}-{\frac {1}{2}}{\bigg )}\,$
	$=\,$	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 1072272882 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 02 May 2026 00:31:42 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {n^{2}-1}{3}}\,$