418531 ภาคต้น 2552/โจทย์ปัญหาความน่าจะเป็น II/เฉลยข้อ 3

ข้อ 1

เราได้ว่า

$\Pr(X=Y)\,$	$=\,$	$\sum _{k=1}^{\infty }\Pr(X=k\wedge Y=k)\,$
	$=\,$	$\sum _{k=1}^{\infty }\Pr(X=k)\Pr(Y=k)\,$
	$=\,$	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 244786981 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:19 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }(1-p)^{1-k}p(1-q)^{1-k}q\,$
	$=\,$	$pq\sum _{k=0}^{\infty }[(1-p)(1-q)]^{k}\,$
	$=\,$	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 169375017 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:20 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 251440187
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:20 GMT


Sensitive client information

IP address: 158.108.32.49

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

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 276403057 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:20 GMT Sensitive client information IP address: 158.108.32.49 $E[\max(X,Y)]\,$

$=\,$

พิจารณา 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 219106185 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:21 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(\max(X,Y)\geq k)\,$	$=\,$	$\Pr(X\geq k\vee Y\geq k)\,$
	$=\,$

ฉะนั้น

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 228344687 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:30 GMT Sensitive client information IP address: 158.108.32.49 $E[\max(X,Y)]\,$	$=\,$	$\sum _{k=1}^{\infty }{\Big [}(1-p)^{k-1}+(1-q)^{k-1}-(1-p)^{k-1}(1-q)^{k-1}{\Big ]}\,$
		${\frac {1}{p}}+{\frac {1}{q}}-{\frac {1}{1-(1-p)(1-q)}}\,$

ข้อ 3

เราได้ว่า

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

ข้อ 4

เราได้ว่า

	$=\,$	$\sum _{k=1}^{\infty }\Pr(\min(X,Y)=k)E[X\ |\ X=k\wedge \min(X,Y)=k]\,$
		$\sum _{k=1}^{\infty }k\Pr(\min(X,Y)=k)\,$
		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 258251924 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:23 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(\min(X,Y)\geq k)\,$
	$=\,$	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 188345604 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:24 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(X\geq k\wedge Y\geq k)\,$
	$=\,$	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 200592800 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:25 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 207308035 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Tue, 14 Apr 2026 10:11:25 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }(1-p)^{k-1}(1-q)^{k-1}\,$
	$=\,$	$\sum _{k=1}^{\infty }[(1-p)(1-q)]^{k-1}\,$
	$=\,$