Question

A) Each month, a manufacturer stocks one warehouse with good x and two warehouses with good y. Monthly sales for x and y (as fractions of a full warehouse) are then random, described by the following joint density:

T1,9) 1-k(x+y), if x € [0, 1], y € (0.2) 10. else

Find k to make f(x, y) a legitimate density function.

B) For the joint density given in part B, find Cov(X, Y ) and Corr(X, Y ).

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Answer 1

2533333333335.23333 3 33333333.5.2 Ans Given fix, y) = $ 1-K(x+y) 2€ [1], y €[0,1] else This is a joint density function in x {y we know that for a function fery) to be a legitimate Enti o density function conditon that LG fex,y) dedy = 1 Here x from 0 to? y from oto2 fery) dx dy-1 2-0 y=0 for everybody = 1 | ) de dy = 1 Sede di dy=1 Sdx= x 150 Y=0 450 450 Jody=y 19 [laz je 1,2)-1

299 - *[ <1+ Beste) du] =1 2- Klit 2017] = 1 - 3K = -1 Hence the value of not so pff becomes f(x,y) = 1 - ļlity) 2€6,1) YE [0, 2] O else B. Cov(xY con (X,Y) wkt for formula cover outra J Soy stosun dediy) - Maky covex jp) is I coulx,y)= y |- Mylly where - Me & My are means of X&Y Hence we have la fx cada mean = Ha= EX]- Р - «у - у 4,

for this we have to find for & freys (marginal densities we have 00 fxcol= | fax,y) dy 4 - Fyry) = 1 fingsde Here ferrys = din fasty) ostel Osy22 else SO 2 fined fr-scrap).dly + Piese spargely = 1977-* (79***13] = (2-0) - } [622-0) +450) . = 2- 28 fue = a[i-1]+4 + 4 MUCH! WN

Ssanlines Simila - (4) Jar) IyM [13+ [ ] - La - ) - 9 - Llo- ) + ) +1 : (- 2 부 - 름 글 : ( So fly = E(X)= (x forces de - (-) dx F ماس ex 스의 | f. = (2) - () MAMA

My = ey)- , yfylgidy 4=6 = 547-5)do = 1 19) - 4 (8) Bo Cov(x,y) = cov (X,Y Try ferry) de dy - Mx My 53 +3/-1-3 109) desdy – (435) 950 y =6

19) - 9146 - 2 - [19)+ (9)] – 25 31- $ [ +4 ] - 28 F1-8-20 Covex,y) = { - -0.01234 cov wkt cor (x,y)=9= cover, y) 6x Gy for we have TE [x+] - porta G”Ety) - chipsJe u de o fertu de

Ес”) - в w 5м . се ( 3) - **(-3)250 - eets/-5), «да 9 Яхь — = 5 9хь L- hz = (2) - (*), з | 1 - 1 1 1 •r (4-x).) сан . В

19) - 9146 - 2 - [19)+ (9)] – 25 31- $ [ +4 ] - 28 F1-8-20 Covex,y) = { - -0.01234 cov wkt cor (x,y)=9= cover, y) 6x Gy for we have TE [x+] - porta G”Ety) - chipsJe u de o fertu de