Suppose X and Y are jointly distributed rancloni variables with joint density function f c(x+y ) 2 0 :Ox,y1 : otherwise — (a) Compute c so that f(x, y) is a valid joint density function. “ [1 3) (b) Compute the probability that X + Y> .5. —I 1 — 2 +-t4) ?(hi ci C) 0 43 - fJ ill 5. - MORE ON NEXT PAGE 7 z 4 — ,/ I Rt/ _( c1( ô - 1) ‘c 4 JJi 3 ‘ - ‘i r’ •:- D \ “5jl.r II + X \jj 1- \SL1’ :-7• -p .ot— c S a) c_—, C . )- O/ -: icr’ I) %jjçp s—) \J’ — Q -d a) - ll CC Ni ÷ N 4 O 5 __,c__— >- >< N + z_____ c1 I’ c_ ‘%\1r H N eJ X - ‘.SjLj- IL .ikj ‘- cj 9th let X denote If a random customer enters the Smiths Grocery on 8th and number of denote Y and er number of peaches purchased by the custom Smiths has the of oranges J)urchased by the customer. The management ing table. estimated that the joint pmf of X and Y is given by the follow x Y p(x,y) 1 2 0 .2 .25 1 .3 .13 2 .1 .02 the difference between (a) Use this joint prnf to compute the probability that s purchased the number of peaches purchased and the number of orange 1. to equai by a random customer is larger than or i (1)2(O)) ô, ) (b) Compute E(X + Y) (.z) + 3) 4- 1- -‘4 (4 4— I • II c ‘I’ V r. CN I (- 1!’% Ir. ‘ft c_ + >.q V / I’. 2/ Q5ZQ Q : / J 5 Ic 5z.o 1)iç&)€ (%)%72 2 c1 — \-- “4 “4 1 r- + ‘N 7 c_v - -ç ft (\ -J cS I L ‘ \\) cc %_____ -< -t 1$ I’ rc, Sd >.-‘ c I’ N _,‘ c. ‘I / U-I ‘I -1- c? , ‘ \-/ cy J\ -z c “ v4 N f1 oO. Ir cV 4 r scJ, - r3 C “ \ (r I r3 (I ‘-‘1 I I i-i $ + Ii j2 4I’J’ \-::-1I II -- N - 4 N 41 —s-.j J%J — hJJ b ‘%f ‘4- C-r’ -- %J I) s < ksLc fK A (Th J1 Il 4 0 11 o1 JDa p II .1 25 is random sample of size 25 from a population whose X , 1 6. Suppose X distribution has probability density function: ..., p(.fX :Ox< — 0 : otIicrw’ise (a) What is the approximate distribution of the sample mean e) = 0 4 0 .4 MORE ON NEXT PAGE (b) Compute the approximate probability that is less than — L 1 ç a) (c) Discuss the validity of the approximation you used in parts (a) and (b). / Jot,LU-cf I rLL3’ C) r. i S 4 44 c:t 1-4. V rJ \pJ N c..J A. flN N It r.1 2’ rzc r k2 ,/t( %or t&&c 4% >1 2.33 233) AJ(o,1 ‘A? ) 1;c( ) ?(- 133( J2( iL 4 3% 2” )) 133 C,,’) -x i3(4 %3) A x ii Ii I,’ N N LI ‘I, 3V 1 L 3CK N LA I JI ‘U _t 4 tJ II 111— lqL -t j) N II h. ‘I c. L’ ‘S 11 4i ‘4- rT 4’ A ii t4%1 V rt I, ‘1 It ‘I 11 -4- 12 1 I—’ p 4 0 I’ H Ii —I U Lr NJ 1- —A 1: 0 1’ V C I’ I II 0 I) a Lj I’ 0— U 1 0 - I * -4 I ‘ 0 I ‘L)