Question

The joint distribution of two continuous random variables X and Y are given by: [xx{xy) = Cry, for OSIS ys 1, and 0 elsewhere a) (2pt) Find C to make fxy(x,y) a valid probability density function. Enter the numerical value of C here: b) (2pt) What should be the correct PDF for x(x); 1. fx (I) = 2r for 0 5r31, and elsewhere. 2. fx(x) = 3-2 for 0 Sis 1 and 0 elsewhere. 3. fx (x) = 4r(1 – ) for Osrs 1, and elsewhere. 4. fx(x) = {(1 - ?) for 0 Sosys 1, and elsewhere. Enter your answer for subquestion b) here (only 1, 2, 3, or 4 is accepted) c) (2pt) What should be the correct PDF for y(); 1. Sy(y) = 2y for 0 Sy 31, and elsewhere. 2. SY(u) = 3y for 0 Sys1, and elsewhere. 3. SY() = 4y(1 - ?) for 0 Sysl, and elsewhere. 4. Sy(y) = 44" for 0 Sysl, and elsewhere. Enter your answer for subquestion c) here (only 1, 2, 3, or 4 is accepted) d) (1pt) Find the covariance of X and Y. Enter your answer here for subquestion d) here (round up to 4 decimal points)

Answer 1

Given joint joint denity function of two cont nuns Ru x and y ie Scny OLNLYLL fxy (n (MY) { o o.W Y=X "-/ (11) X (4)

f(x,y) no ing paf ff f(ny) dy dna If any dy dn IL 2. A dnt All-x2 dn at 2 를 2 - Dan 1 ㅈ 르 IMA 2 C 들 xas X 르라

b ) marginal pdf of x 1e fa f fin, y dy y I frye dy + 7 Z 8nly 2 2 fal 4444=14 Tu. To 05 x elif of le f(x) 42 (1-x2 ochLL Mio e rel 3. option will be correct marginal pdf of y yre. f(1) = Sf(n. 4 da R 9 6 s snyda 896 49.92 If(y) = 443 ocy 5.) 2 3

+पये L [-९ 4th option will be correct a) EPTE A finis dydu. say.&ny diy da न Hydr कर (1-3)de (x-f)तर 3 3 M 8 3 8 6. Z 17 2 - 6 4- xx K 3

مه - F(x)= I a. fM) du a. 400 1-2 da st 4. S (x²n4) de 4 (x3 لا 7 c 13+ *) 4 حي (x) = LI وله (6} . و ال = (e c 9. 17 dy ) والی او له = 4 gs در ؟ - FU1) = $ 5

so au exiy)<E XY) - E(X). Ely) 4 8 x 4 15 5 - on = 0.4444 -0.4267 >> cou(x, y) = 0 0.0177