Question

Let the joint density function of random variables X and Y be f(x,y) = 8 - x - y) for 0 < x < 2, 2 < y < 4 0 elsewhere Find : (1) P(X + Y <3) (11) P(Y<3 | X>1) (111) Var(Y | x = 1)

Answer 1

Ginen the joint density function of random variables X and Y , 24: fex,y) = ŞI (6-7-9), Ocxc2, 1 0 qrow GB 2 3 aty : : t f ö a xty3, 922 I shaded region, å ... 1 P(X+Y <3) .. 1 =SS fox,y) dy dx . . and ĭ 3-x op i tad (6-x-42 dy dr ary (3 x=0 y=2 3-2 * Cocorregut com 3 do OL

- *(1-x) - ļ (--3-4)] dx .-x+x² - // (9+x²6x-4)] dx ,Š6-7****** ***32]dx $$ (3.-4x+z14 dx - * (?(22!– 4 (23! +*2 (23] 2*176-4+%] : =% [ Part Ye Da quş. Geen @ P(463\x71) = P(423,871). ( By the definition P(x>1) of Conditional probability )

هو بهم و « (۶۶۱ ۷ )۲ et shaded regions دو || (771 (79)م در my • j f feesy) dedy si & 6-x-y)dz dj - وه [ ع ف - ی) - ش به [د - (12) - وار وه (- - ا ء[تر ، =

16-- + (9-4) * 2- ] - g /4+0.25 - P(x+1)= Ŝ fxix) dx, fact) is the marginal density function of x. Marginal of x, fxcx)= if fexsy dy : - $'(6-2-y) dy = [069"-269.443] [6x2-x+2 --4 (1-42] (12-24-6) - 5 (6-2x] fy(x) Some . OCLC2 3. P(421)-456-x) dx 3713(25?- 1623"] **(3-4(4-)-. (Va) 0.375

...P(43)*71), % Do V (Y/xy), Elixaby(€ ly\x41) | f(\\x-x) =) for guien ocacz, feyla) - feasy) 1816-x-4) fx (x) = 1/4 (3-2) =/(0-x-4), zey24 3-7 1 ) - for given ocae2, the conditional density of (Ylx=x) is flelx-x) = 1 ( 6-x-9), 224 c4 - -- - -

Now fly1x1) = ₃ ( 5 ) = 1 (5-4), zeyen €(1801). y feylami) dy - Jy 4 (5-y) dy o t ( By y3 dy 34 (5624-343.4] -(?(16-4) - (ou-0)] * # ( 30 - ] +(99) € (p<(xv)- Sp* fcylacody =g2 + (54)dy - 11892.99dy -+(362-462") et (Tou-s) - (256-16]] - [ 50x5–60] -4 (280_60] -

sler) () 8-3333- 6.02 78 ’ -30SS ,