Question

Two docnete fo Random varrable X and Y have the following joint pre fro,y)= {k (2x²y + 2y x² - 1 x = 0, 1, 2 Dothonorge 9 = 1,3 ① bay = Cor(xy) (1) Use Mct,, tz) to check of your answer to (1) is correct

Answer 1

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lets first • find K 2 는 Z 20 1,3 ET) . 2 ㅋ 세ke4월인가 그 극 I y리, 3 M20 K 및 -h+ huz) . नाम J ㄷ 2x +3) + 2x IF) -2 M ㅋ ka 채 도도 대령과 5명 x2) 이 (2 (F5) * x)) a 8에를리의 + Strate) at 18) 다 ( bale) +12) +13) = ) 2. ET. K 8 키 0 2 ㅋ Ga 기0 2. 기 48

2 !. f (x,y) = I (2x²y + 2y x²-1), X20, 2 y=t, 3 48 o otherwise л covery) E(KY) – E(X) E(3). Elxs 2 II n. f(ay) ny ut ( 27 (272 y + 2xy = -x] ( 8x² + 8x 3 20 ti 18 8 - 20)7. 8 (0+1+8) + 8-(0+1+2) – 2x [0+1+8) -2 (0+1+2) s fbog +603] 18 2 z 48 54+18=72 ५४ 48 +2 3 EX) 22 8 2 O

4) и Е(7) TLE (217° ++ 2y-у-у-2) [ + (от +10 - 4**-*)) 16 «(+1+ч ) + 163 : £(1) 216 [] = = . – 48 48 а) 9) SE(x) = [[ny. f(x,y) ny 48 + 2xy* - у - 19 22 ( 21 ) 2 г Т. 21 ( 1s 2 + авч — 42-4» ч") (2or(en) + 1С: (ант) -ка (онн) а толер + -36]= 4. – 48 .: (81) - Тао

using , 6 8 6 we get, Corc x, y) = ECXY) - E(X) E (*) & 3 x 8 3 phen 2 Cov(X,Y cuenta O Ans Ô t, X+ tzy Ele Mlt, tz) I ny hy) f(18) tin tzy IIe tr ty tx ty e 2x²y + 2ye e EI I 48 a tzy txt, -x e e -le tx, tz. 3t21 tx, tz 3t2 e +3e te (2e + - 1 E 2x the be n to 3t2) xe. e 69) + *% ity) 48 trise txtz 3 tz) ete

(23 re 48 a tuttz +6n2 e tinttz 1² e + 2e txt 3 tz tixttz tattz t be text3t2 tinttz tut3 tz a e e - e El - tuttz 2 tint3tz tinttz a e +5ne +5e tattz) 48 a to to 3tz tz t, 26) ti le 48 tz 3 tz 도 2t, e (e the + sé etue) 48 télite" tett) + se cite te --M{t, th]z + the teet se 3tr + (itetit te sterse Nowy ECXY) = 3 2 la at, atz It-o, to 2 om Ltı, tu) = 2 they fet? +15e!!) the (itetnte''){et24152) 21+ le trtise Mitigt) = 362) (2e toit loe? alt, atz I le the at, ata at, 48 e ati stz ty 't se ti It ze tse 2t)

ElxY) = 2? M(t, tz) at, atz to, t₂ = 6 O I 48 ( ét? + 15ae) (2e 2e Flue 9). + 16.12 48 Hence Now, El XY) is verified to be correct. E(X) = 2 m (tuo) 어 at, t=0 2t, 2 Gal titetitse 78 t-o o + 2 a(zeº+toz 8 co Z EX is abo verified to be correct Similarly, ECY)= M(0, te) Itino. Itulemusely to at216 t=0

- Śle'tisne") 16 2 Wlao 6 ... E(X4)- ELX) ECY) 4-12 20.