Question

Please answer Problem 18

Problem 17 Find the constant k that makes the following functions PDFs. (a) p(x) = k sin x, 0 < x < a (b) p(x) = kx2(x - 1)2,0 < x < 1 (c) p(x) = kx(1 – x)}, 0 < x < 1 (d) p(x) = k, -1 < x <3 (e) p(x) = kx'e-3, x > 0 Problem 18 For the PDFs in Problem 17, compute the expectations, variances and standard deviations or their associated random variables.

Answer 1

. Plala ksina. OLACT IT Plar dar k sina da ak sina da k ort) - cor (]} k (Coso - Corcm) K (1-(-1)) = 26 1 a sina del 1 ( 2 ราก a da Facord -cora da - ha cosa]." - 5- cosa de at haxcosx)" - (asinalo? les cas cm)) -((-sin(T) + sincoj]} IT -(0)+ o))

(x) = x² sina da х” со cora +2 asinant о соѕt ) гсоѕСП) + 9 socTT о со СП) - + con(s - т — 1 с-1 42 Г (0) + (-1) - TE . (п+91T - - 2 ) от (пат —4) Nix) = et(*) -(E(кој 2) ( П', эп-ч) — * - T2 T-- Ta - TT 2 + — 6 SiD-tv (x) - P( 2 ) : 22- ( x-12- 12 + Т- 2 о ст (ь) късная * (? - x 1) qя

=kS (29_22² +22) da ( - 1 + 13) -- K30 - -- = k=30] E(X)= 301 1360-0321x = 30 = 30 (23 (222x + 1) da 20 Sof-224+28)dx 230 (90 - 2013 ) - 30 (1 - 2 + +) = = = = 22plp = 30/+28+30) 120

E(x) = 5 зохчC-12 ах во эч (o 2 +244 во ( 2x+ЧЈda зо ( - а .х). ИИ ИЗРААР от зо ( - чо +41) \os vcx) - E (х2) - (Ecx)] 1 - - jno — los los 120-los uno 15 42 о О - по со yuno Р(а): k (Ac1- ) , olx 1 (ка (( - - 31 (1-a)) ax ря (1-12 2х + 352) 4 Ek (1 - 333°43 ),

Д ВА J. 3 - - и с - - 1 ( 1 - +, 2 ) - Бо ЕН----------- - 2010) Е (x) = no a fə - яч — 31° 432 ) 4x ело (а - - 329 -3° ) dx с 20 ( ), S о ч. 1 - ) - 06 ( - до +a ) no

c. 20 (o) - 2 Carau_3424323) dx 0 1 0 - - 345+ 329. RAPPPP22 - 20 ( 15.03 + -286 +315 630 630 630 Vox - suo 630 g 4230 0.7460 S670 + SoD-Jo.2460. -1ca53 3 - € Pla)= K -1<253. sprais da = 2 k[al 7 K (3+1)=1

to be E(82) = - (22+1 - 28 = 14 V (X) = E(X²) - (E(X)] lu-6 spalvelut

Pla I Kaß o ano kaBo / da el de .-2/2 da Ebatan da K . 96 as 24 en da UX 3X2

E(x2)= to Syedx I 120 x64 7680 W(X) = 7.680 - (768) LUULDU 96 80 - 64 = v(x) S.D= 16 Jucxo = √16 = 4