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 245498052 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:37 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(X=Y)\,$	$=\,$	$\sum _{k=1}^{\infty }\Pr(X=k\wedge Y=k)\,$
	$=\,$	$\sum _{k=1}^{\infty }\Pr(X=k)\Pr(Y=k)\,$
	$=\,$	$\sum _{k=1}^{\infty }(1-p)^{1-k}p(1-q)^{1-k}q\,$
	$=\,$	$pq\sum _{k=0}^{\infty }[(1-p)(1-q)]^{k}\,$
	$=\,$	${\frac {pq}{1-(1-p)(1-q)}}\,$

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 238685720
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:40 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 229774260 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:41 GMT Sensitive client information IP address: 158.108.32.49 $E[\max(X,Y)]\,$

$=\,$

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 221555379 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:41 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(\max(X,Y)\geq k)\,$

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

$\Pr(\max(X,Y)\geq k)\,$	$=\,$	$\Pr(X\geq k\vee Y\geq k)\,$
	$=\,$	$\Pr(X\geq k)+\Pr(Y\geq k)-\Pr(X\geq k\wedge Y\geq k)\,$
	$=\,$	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 219033516 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:43 GMT Sensitive client information IP address: 158.108.32.49 $(1-p)^{k-1}+(1-q)^{k-1}-(1-p)^{k-1}(1-q)^{k-1}\,$

ฉะนั้น

$E[\max(X,Y)]\,$	$=\,$	$\sum _{k=1}^{\infty }{\Big [}(1-p)^{k-1}+(1-q)^{k-1}-(1-p)^{k-1}(1-q)^{k-1}{\Big ]}\,$
	$=\,$	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 228432029 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:44 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 215721930 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:45 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

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 163917676 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:45 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(\min(X,Y)=k)\,$	$=\,$	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 232788762 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:46 GMT Sensitive client information IP address: 158.108.32.49 $\Pr(X=k\wedge Y>k)+\Pr(Y=k\wedge X>k)+\Pr(X=k\wedge Y=k)\,$
	$=\,$	$\Pr(X=k)\Pr(Y>k)+\Pr(X>k)\Pr(Y=k)+\Pr(X=k)\Pr(Y=k)\,$
	$=\,$	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 214088526 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:47 GMT Sensitive client information IP address: 158.108.32.49 $(1-p)^{k-1}p(1-q)^{k}+(1-p)^{k}(1-q)^{k-1}q+(1-p)^{k-1}p(1-q)^{k-1}q\,$
	$=\,$	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 220605104 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:48 GMT Sensitive client information IP address: 158.108.32.49 $(1-p)^{k-1}(1-q)^{k-1}{\Big [}p(1-q)+(1-p)q+pq{\Big ]}\,$
	$=\,$	$(1-p)^{k-1}(1-q)^{k-1}(p+q-pq)\,$

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 207733276 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:49 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 217821088 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:49 GMT Sensitive client information IP address: 158.108.32.49 $E[X\ |\ \min(X,Y)=X]\,$
	$=\,$	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 203049936 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:50 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(\min(X,Y)=k)E[X\ |\ \min(X,Y)=X\wedge \min(X,Y)=k]\,$
	$=\,$
	$=\,$	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 230266847 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:02:59 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }k\Pr(\min(X,Y)=k)\,$
	$=\,$	$E[\min(X,Y)]\,$
	$=\,$	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 245465407 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Mon, 16 Mar 2026 22:03:00 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }\Pr(\min(X,Y)\geq k)\,$
	$=\,$
	$=\,$
	$=\,$	$\sum _{k=1}^{\infty }(1-p)^{k-1}(1-q)^{k-1}\,$
	$=\,$
	$=\,$