418531 ภาคต้น 2552/โจทย์ปัญหาความน่าจะเป็น II/เฉลยข้อ 3

ข้อ 1

เราได้ว่า

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

ข้อ 2

เนื่องจาก เป็น random variable ที่มีค่าไม่เป็นลบ

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

$=\,$

พิจารณา $\Pr(\max(X,Y)\geq k)\,$ เราได้ว่า

	$=\,$
	$=\,$
	$=\,$	$(1-p)^{k-1}+(1-q)^{k-1}-(1-p)^{k-1}(1-q)^{k-1}\,$

ฉะนั้น

	$=\,$	$\sum _{k=1}^{\infty }{\Big [}(1-p)^{k-1}+(1-q)^{k-1}-(1-p)^{k-1}(1-q)^{k-1}{\Big ]}\,$
	$=\,$
	$=\,$	${\frac {1}{p}}+{\frac {1}{q}}-{\frac {1}{1-(1-p)(1-q)}}\,$

ข้อ 3

เราได้ว่า

$\Pr(\min(X,Y)=k)\,$		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 442626177 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 12:19:17 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(X=k\wedge Y>k)+\Pr(Y=k\wedge X>k)+\Pr(X=k\wedge Y=k)\,$
	$=\,$
	$=\,$
	$=\,$
	$=\,$

ข้อ 4

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 489852536 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 12:19:19 GMT Sensitive client information IP address: 158.108.32.49 $E[X\ |\ X\leq Y]\,$	$=\,$
		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 463552860 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Wed, 25 Feb 2026 12:19:20 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(\min(X,Y)=k)E[X\ |\ \min(X,Y)=X\wedge \min(X,Y)=k]\,$
	$=\,$
	$=\,$	$\sum _{k=1}^{\infty }\Pr(X\geq k)\Pr(Y\geq k)\,$
	$=\,$	$\sum _{k=1}^{\infty }(1-p)^{k-1}(1-q)^{k-1}\,$
	$=\,$	$\sum _{k=1}^{\infty }[(1-p)(1-q)]^{k-1}\,$
	$=\,$	${\frac {1}{1-(1-p)(1-q)}}={\frac {1}{p+q-pq}}\,$