Question

explain as much as possible, thanks!!

Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 po. Suppose the actual air pressure in each tire is a random variable for the right time and for the left with pdf given below. FX,Y) - K. 22 55 32 22 y 5 32 otherwise (a) Compute the covariance between X and Y. (Round your answer to four decimal places) Covex, (b) Compute the correlation coefficient p for this X and Y. (Round your answer to four decimal places) P

just in case the other picture was not clear

Answer 1

Note: As the pdf is symmetric in x and y (i.e we can swap x and y with each they have same form and range) so the results computed specifically for X hold for Y too, like E(X), Var(X) are all same to E(Y) and Var(Y) also X and Y will have the same Marginal PDFs, hence they are identical(not independent though!!)

f(x,y) = K(x² + y2) 22 su 2 3 2 2 22 2y 232 otherwise To find I we'll solve: 32 32 SS f(x,y) dady = 1 12 22

32.32. => n Texty₂) dn dy 1. 22 22 32 32 3.2.3.2 1 ke SS I a? dady tk y z dndy 22 22 22 32 32 3 2K ss u² dudy = 1 22 22 27 2k (32²-223) , to 1 3 - 6 2 t = 6.78119 x 10 3 20x(329-223)

3. ELxx) ffre К у 192) 1 жач 2. п К ) "dy d). . 7 пл - 4.2. 2 2.И. . dn ли --- ки" 4 1; to 2 }(ingdong 91 ,, . :. ELх*). f(x). « км (+2°) 44 45 dy t k ff ny dxdy 3. 2. ГЧ La7 Ч 1. 2 2 пқ 321-22 к 2. 4 2 ди о г 4. 1L - ж[23732 4. а. x to the = lo к 611-221) Екк (12-1") - 412-12) и 6

Similarly + Ely) - Elx ) = 10 u (323 224) ☆ (32?- 22-) (32222) to h (32 ² 2 2 2 2722) 6 de 2 22") {se 625 +*«(32-12") 3 27.305 13 9 ELY) CO2- Elx) 2 27.305139 2 745.4729) .: Coul x,y) = E(XY) - ECX DECY) Ⓡeseeseen Cor(X,Y) Enreoge -0.0927- Now? E (x²) = x² (nty 2) andy. C. 2 K a SS d 32 ka 5x5 10 tk a lo 22 22 a usanduda t affory dedy. Le sue CATE] = 2toak (325_223) + ka (32²- 223² E753-8489). 2. E(Y2) = E(+2) = 753-8489 %. 2 var (X) = E(X²) - (E(x)) = 8.278294 9

CornCX,Y)= Coulx, y Var (xlvas (9) w 2 e vorlx) .. (or (X,Y) var (x) evol 0.01119 0.0112