Question

2. [x] Suppose that Y1, Y2, Y3 denote a random sample from an exponential distribution whose pdf and cdf are given by f(y) = (1/0)e¬y/® and F(y) =1 – e-y/0, 0 > 0. It is also known that E[Y;] = 0. ', y > 0, respectively, with some unknown (a) Let X = min{Y1,Y2, Y3}. Show that X has pdf given by f(æ) = (3/0)e-3y/º. Start by thinking about 1- F(x) = Pr(min{Y1,Y2, Y3} > x) = Pr(Y1 > x, Y2 > x, Y3 > x) and use the technique similar to the one for the uniform distribution example. (b) Consider the following five estimators of 0: Yı + Y2 Ô2 = Ô, = X, Ô, - Yi + Yo + ¥½ Yı + 2Y2 Ôz = Ô4 = X, Ô5 Ô1 = Y1, %3D 3 Which of these estimators are unbiased? Among the unbiased estimators, find the one that has the smallest variance.

Answer 1

Page 1 I suppose 41, 42, 43 denote nondom Sample from exponential distribution whoose p.d.f and e.dif are given by f(y)2 1 2 0 H10 & Fly)z 1-es , y 2o, respectively with some unknown oro. It is also known that & ril=0 @ Let, x= ming 4, 42, 43}. show that x has p.dif given by fcay = (310) 03/02 solution : Let, x= min $4, 42, 43} z Yu , where you = Smallest ondeen Statistic & Y = gth onden statistic cleanly, Yo = 4 (2) 2 Y . ا ردعم ,e 3 / 4 & edd, F(y) , -ē 910 Now, the codid of xis Fx Cam = Fy (L) - P(klu (R) = 1- P(1 x) 1-P (Smallest Yi 7x) (sime, min Yi XX >1-Fins = 1- p [all Ye >] ( aul Li YR --P [ 4. XX, 42>X, 43 x2] = 1- P(UTK). P(1272) • P(432) Hene, 11, 42, Yz are the granden somple se, p[Y> X, Y2> %, 43»x] = P(Y.SR).p(\:>>) •PCY;>x) So, 1- FOy = 1- - PCY, 5x)]-[- P(9232)][i-P(4359 = 1-[ 1 - Fus]. [1-F2 (x)]. El File 7

pase: 2 Home - YoYL, 43 an YoYL, Y3 are the handom • exponential distribution having g. dd fy (Y)2 1 - 0 apice the & handom vaniables y, Y2, 73 and chcependent and having c.ad Fyilda 1-2 310 So E -no row, fun we set, - Fax cay = 1- [1-(1-ē 40)]: -(1-2 *b)]. [1-(17may 7 1- Faly al- é 3x10... To get respect tox pdf at y . we will diffenciante it with Now, diffenerating a with respect - 1 Focus}= 0 - 3 2 310 to a weget - Sx Casa de l Socha £o cas] / >> Sx casžē340 ex kay pidid, har when, x= min (t., 42, 43), ie, & has minimum vahe of Yi , 121(1)3. So, we can write the p.dit of xis Sx 3 e 310, Pro) Ans) co, y follows exponential distribuline with Rake, panometen &

Pas 3 NOW, EC) = 52. 3 e 340 du 2 ne 318 de taxes, 3x = 2 3 0 - uu wo wboot (2 psim, s nepamidra ring > ny)! 7 who wk and E (x) = 1 x fonda 2 $°r= NVO da J a 2² e ² edz, tave, 34 22 2 FC3) 13, Zz = ren Now, v(x)= E(X²) - E(x))?

payz 4 Here, Geiven that, in te 210 and Ecuza Now, E(F) y te loody al que lo dy , ty tarce, y 10=2 so (6.21e? od 2 د لاله زد 8. = to o for an er da 2 on see? 234 dz 204T (3) 2202 So, VCY") 2 € (4²) - Erry J² 220²020?.... Question & I consider the following five estimation . 2 =4, 62 = Yith a = 414242 Og = x, osa Yinfyty 3 which of these estimaton one unbiased? Among 1. unbiased estimatory find the one that has the smalles vanane. sol terte, Yo Ya and Y2 are hendom sample fue y & bf has meano, vaniare 200 (3 EC4/2 of vCY) 2202 ( Sines W 2202 ( sina, wines Soy Elli)= 0 & U(Y)2oL a & Vly.) zon a , it isl; 2, 3. - Now, "E(6) E (4) = 0 (using o n is is ansiased estimate it o.

pasens 2 ² [E(4) + E (42)] (sines we know Crithemont Yn) 21 [oto] EchiD ) 20 unsiased estimade of o. I E (03) & ( 12+242) - [ECYD) +2e crey) Ś [0+ 20) i sm, eccy Jee ECU) ) non 0 (using @ So, 08 is unbined estimation on ou r E (64) 2 E(X) = AS, E(0)79 Se, en is niet un biased Estimation of a. . O E Cês) = (x + y 244 ) BLE (4.97 12 (47) + EC4917 25 [048+ E) - Q do, is is unbiased Estimation at o. Hene, di, 2 oz & os are the unbiased estimatondo. Smallest vanicene Amory all unbiased petinchons . vôo.) z v (41) 202 [usty ] ... a (62) av (Yit 4 ) a uf [v (Mig+ V(42) + 2 couco a clu(x) ve)

Page ²6 B o sa I 10402] 20 min. (sta, As 41, he are indeportach 3o, cerch ) zo) 3 • VCO3)2 V (+242) [v(41) + V (242) +200 (41,244) 7 = $ [-41)+ qu(YX) fcev ( 41, 407 = 3 [or+ 40 =0) { Stre, cov (154L) = 0) Asi oaax is not unbiased estimation. So, we will not considen this @vcoś so y tiene su Yety) 2 + ((2x) + (42) +v(43)+ ab cov 41,437 = 1 0240702) (cov (4,4;)=0, titi os Yi's are independant cz1(1) 3, j = 1(!)} av de, (Os) su(ta) vl e) < vlod Here, the chestimate os = 91 +42 +42 has Smallest variance amony at the unbiased estimaton on an +4) Os. 3 hay