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 831565044 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:55 GMT Sensitive client information IP address: 158.108.32.49 $=\,$
	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 820367100 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:51 GMT Sensitive client information IP address: 158.108.32.49 $=\,$	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 812831866 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:51 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 820560291
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:51 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 819908277 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:52 GMT Sensitive client information IP address: 158.108.32.49 $E[\max(X,Y)]\,$

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 851021474
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:52 GMT


Sensitive client information

IP address: 158.108.32.49

$\Pr(\max(X,Y)\geq k)\,$ เราได้ว่า

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 851577473 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:52 GMT Sensitive client information IP address: 158.108.32.49 $=\,$	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 819414836 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:52 GMT Sensitive client information IP address: 158.108.32.49 $\sum _{k=1}^{\infty }{\Big [}(1-p)^{k-1}+(1-q)^{k-1}-(1-p)^{k-1}(1-q)^{k-1}{\Big ]}\,$

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 845582181 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:53 GMT Sensitive client information IP address: 158.108.32.49 $=\,$

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 814272543 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sat, 31 Oct 2026 06:17:53 GMT Sensitive client information IP address: 158.108.32.49 $E[X\ |\ \min(X,Y)=X]\,$