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 939930731 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:44:55 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 959021845 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:44:56 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(X=k\wedge Y=k)\,$
	$=\,$	$\sum _{k=1}^{\infty }\Pr(X=k)\Pr(Y=k)\,$
	$=\,$	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 939995739 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:44:57 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }(1-p)^{1-k}p(1-q)^{1-k}q\,$
	$=\,$	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 882696334 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:44:57 GMT Sensitive client information IP address: 158.108.32.49 $pq\sum _{k=0}^{\infty }[(1-p)(1-q)]^{k}\,$

ข้อ 2

เนื่องจาก 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 962102151 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:45:08 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(\max(X,Y)\geq k)\,$

พิจารณา

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 897311133
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:44:58 GMT


Sensitive client information

IP address: 158.108.32.49

$\Pr(\max(X,Y)\geq k)\,$ 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 885448281 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:44:58 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(\max(X,Y)\geq k)\,$	$=\,$	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 930331350 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:44:58 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

		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 959841149 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:45:16 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }(1-p)^{k-1}+\sum _{k=1}^{\infty }(1-q)^{k-1}-\sum _{k=1}^{\infty }(1-p)^{k-1}(1-q)^{k-1}\,$
		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 899208802 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:44:59 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {1}{p}}+{\frac {1}{q}}-{\frac {1}{1-(1-p)(1-q)}}\,$

ข้อ 3

เราได้ว่า

		$\Pr(X=k\wedge Y>k)+\Pr(Y=k\wedge X>k)+\Pr(X=k\wedge Y=k)\,$
		$(1-p)^{k-1}(1-q)^{k-1}(p+q-pq)\,$

ข้อ 4

เราได้ว่า

$E[X\ |\ X\leq Y]\,$	$=\,$
	$=\,$
		$\sum _{k=1}^{\infty }k\Pr(\min(X,Y)=k)\,$
	$=\,$	$E[\min(X,Y)]\,$
	$=\,$	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 956925575 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:45:01 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 890229132 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:45:02 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 956762598 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:45:03 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 746382258 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:45:03 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 951360128 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Thu, 09 Jul 2026 05:45:04 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }[(1-p)(1-q)]^{k-1}\,$
	$=\,$