418531 ภาคต้น 2552/โจทย์ปัญหาความน่าจะเป็น II/เฉลยข้อ 3

ข้อ 1

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 550909870 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 26 Apr 2026 13:45:19 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(X=Y)\,$	$=\,$	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 537769674 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 26 Apr 2026 13:45:20 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(X=k\wedge Y=k)\,$
	$=\,$	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 617220118 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 26 Apr 2026 13:45:21 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(X=k)\Pr(Y=k)\,$
	$=\,$	$\sum _{k=1}^{\infty }(1-p)^{1-k}p(1-q)^{1-k}q\,$
	$=\,$	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 562246462 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 26 Apr 2026 13:45:23 GMT Sensitive client information IP address: 158.108.32.49 $pq\sum _{k=0}^{\infty }[(1-p)(1-q)]^{k}\,$
	$=\,$	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 602981319 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 26 Apr 2026 13:45:24 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {pq}{1-(1-p)(1-q)}}\,$

ข้อ 2

เนื่องจาก

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 608677218
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 26 Apr 2026 13:45:25 GMT


Sensitive client information

IP address: 158.108.32.49

$\max(X,Y)\,$ เป็น random variable ที่มีค่าไม่เป็นลบ

$E[\max(X,Y)]\,$

พิจารณา เราได้ว่า

	$=\,$
	$=\,$

ฉะนั้น

$E[\max(X,Y)]\,$
		$\sum _{k=1}^{\infty }(1-p)^{k-1}+\sum _{k=1}^{\infty }(1-q)^{k-1}-\sum _{k=1}^{\infty }(1-p)^{k-1}(1-q)^{k-1}\,$

ข้อ 3

เราได้ว่า

	$=\,$
	$=\,$
	$=\,$

ข้อ 4

เราได้ว่า

	$=\,$
	$=\,$	$\sum _{k=1}^{\infty }\Pr(\min(X,Y)=k)E[X\ |\ \min(X,Y)=X\wedge \min(X,Y)=k]\,$
	$=\,$
	$=\,$
		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 483412232 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 26 Apr 2026 13:46:16 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(X\geq k)\Pr(Y\geq k)\,$
	$=\,$
	$=\,$	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 599669837 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 26 Apr 2026 13:45:31 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {1}{1-(1-p)(1-q)}}={\frac {1}{p+q-pq}}\,$