418531 ภาคต้น 2552/โจทย์ปัญหาความน่าจะเป็น II/เฉลยข้อ 3

ข้อ 1

เราได้ว่า

$\Pr(X=Y)\,$	$=\,$
		$\sum _{k=1}^{\infty }\Pr(X=k)\Pr(Y=k)\,$
		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 909808293 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 23 Apr 2026 21:04:32 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }(1-p)^{1-k}p(1-q)^{1-k}q\,$
	$=\,$	$pq\sum _{k=0}^{\infty }[(1-p)(1-q)]^{k}\,$
	$=\,$

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 877096402
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 23 Apr 2026 21:04:33 GMT


Sensitive client information

IP address: 158.108.32.49

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

$=\,$

$\sum _{k=1}^{\infty }\Pr(\max(X,Y)\geq k)\,$

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

$\Pr(\max(X,Y)\geq k)\,$		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 912363989 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 23 Apr 2026 21:04:35 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(X\geq k\vee Y\geq k)\,$
	$=\,$	$\Pr(X\geq k)+\Pr(Y\geq k)-\Pr(X\geq k\wedge Y\geq k)\,$
	$=\,$

ฉะนั้น

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

ข้อ 3

เราได้ว่า

$\Pr(\min(X,Y)=k)\,$		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 890482703 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 23 Apr 2026 21:04:37 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(X=k\wedge Y>k)+\Pr(Y=k\wedge X>k)+\Pr(X=k\wedge Y=k)\,$
	$=\,$
		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 875849832 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 23 Apr 2026 21:04:38 GMT Sensitive client information IP address: 158.108.32.49 $(1-p)^{k-1}p(1-q)^{k}+(1-p)^{k}(1-q)^{k-1}q+(1-p)^{k-1}p(1-q)^{k-1}q\,$
	$=\,$	$(1-p)^{k-1}(1-q)^{k-1}{\Big [}p(1-q)+(1-p)q+pq{\Big ]}\,$
		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 915966186 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 23 Apr 2026 21:04:39 GMT Sensitive client information IP address: 158.108.32.49 $(1-p)^{k-1}(1-q)^{k-1}(p+q-pq)\,$

ข้อ 4

เราได้ว่า

	$=\,$
	$=\,$	$\sum _{k=1}^{\infty }\Pr(\min(X,Y)=k)E[X\ |\ \min(X,Y)=X\wedge \min(X,Y)=k]\,$
	$=\,$	$\sum _{k=1}^{\infty }\Pr(\min(X,Y)=k)E[X\ |\ X=k\wedge \min(X,Y)=k]\,$
	$=\,$	$\sum _{k=1}^{\infty }k\Pr(\min(X,Y)=k)\,$
	$=\,$	$E[\min(X,Y)]\,$
	$=\,$
	$=\,$	$\sum _{k=1}^{\infty }\Pr(X\geq k\wedge Y\geq k)\,$
	$=\,$
	$=\,$	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 920453736 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 23 Apr 2026 21:04:41 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }(1-p)^{k-1}(1-q)^{k-1}\,$
	$=\,$
	$=\,$	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 898351247 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 23 Apr 2026 21:04:41 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {1}{1-(1-p)(1-q)}}={\frac {1}{p+q-pq}}\,$