Question

statistics and probability

Each front tire on certain type of vehicle is supposed to be filled to a pressure of 26 psi. If the actual air pressure in each tire is a random variable and X is referred to as the right tire and Y as the left tire, with a joint pdf SK(x2 + y2) 20 SX 30,20 Sys 30 otherwise f(x,y) = 0 Where K = 2000 and for all parts require detailed calculations. a. Determine the (marginal) distribution of air pressure in the left tire alone. b. Compute the covariance between X and Y. (Hint: E(Y) = E(X)) c. Explain the steps required to compute the correlation coefficient p for X and Y then perform the calculations. Given that E(Y2) = E(X) = 649.825. (Hint: V(Y) = V(X))

Answer 1

Fir, y) = (hty : 205 x 2 go, 2054 280 , otherwise . 1 o Where k= 3 380,000 380. a) Given X Y: Right Left tire tire. Here we have asked to find of air pressure in left tire le marginal distribution of marginal distribution alone y. Marginal e.dif of Y : 30 f (y) = ( free, y) do 20 2024 230 30 - 2012 + y² da 20 en = k/(x² + y) da k 20 30 (andat (y² da do ER (9 2020.04.23 01:18

sk (68029= 100)) +y? (90-20) Ek [+ [19000] + 10 ye? = 19000k, 10 yek = 19000x Toyxon 190ook 380,000 Lt 30 42 2054230 ta (Y) = 1 20 380.000 This is morginal p.dif of Hon y (left tire) (6) Cor (x, y) = E(XY) - E(X) E(Y) ELX Y = my fire, y) dordy 30 30 ay K (n²7 y dyda 30 30 Hy katy dy dar ay) 'dy da 2020.04.23 01:20

Date Page Ek Špe (900400) + .. ( 909_10")] du? T250 x + 1625oon 30 20 = K sr 250x3 dae + T 1625oon dal 90 30 30 162500 H 162500 =* < 250 (80 *= 40") + 162500 (500)} 250 2 4 - -KS 40625000 + 40625000 81250000 x 3 ) 380000 E(XY)= 641, 4:474 30 Here E(Y) = E(X) 80 - E(X) = (hef (2) dx = y faly) dy = ELY) 30 ECTES:4 ( to toyo) dy 880000 30 36 y dy til 30 J 380,000 43 dy 2020.04.23 01:21

(- vo) + Bo ( - ) 380,000 = E(Y) = 12.5 + 12.8.289 25.3289. cov CXIY) = E(XY)- (ECU)72 = 641.4474 (2513289) COV(X,Y)= -0.1082 10 correlation coefficient e = Coulx, 4) Svarcx). Vat(Y) Here v(Y) = V(x)... Givent E (42) = E(y² = 649.825 verly) - E(13 - [E(y)] - 649.925 - (25.32797 Varly) = 8.2718 .. var (x) = 8,2718 . = .0.1082 com. coeff (X,Y), P (var (4))² -0.1082 8.2718 - 0.0131 2020.04.23 01:22 - -

classmate Date Page There is negative and Y (left tire). correlation between x (right tive) TRR 2020.04.23 01:24