Question

10. (10 points) A function f : R2 + R is called a probability density function on D CR if (6) f(, y) 0 for all (x, y) E D and (i) SD. f(x,y)dA= 1. ſk(1 – 22 – y2) 22 + y2 <1 (a) For what constant k is the function f(z,y) a prob- 12 + y2 > 1 ability density function? Note that D= {(1, Y) ER? : x2 + y² <1}, the closed unit disk in R2 k x2 + y2 <r2 (b) If f (1,y) = is a probability density function what must be the 0 12 + y2 > 72 value of k?

Answer 1

solution :- 0 be a (a) f(x,y) K(1-x2 - y2) , x?zy? 21. x2 + y2 > If f(x, y) will pou balita density function (i f. cn, Y) > 0 a, y E A f(x,y) dA=1 then for all D

Here {x,y) E R² عطر کی *+7221}. x2 + y²=82 let 2= r cosa. y rsino: where ryo. را 2 1 5 6 3 2 1 and I < 2 7231 => o <rs . D = (0,0): osrel, - LOL را 7 (Ne) and dA= rdri do *(1-82) & drdo. - Glov . 5$ t60,37 da a ker facro do K.[12] E (8-83) ar do 1 - 2 CSIV -> do al PEN 2 k 6-4) db = 1 k ㅋ 1. I K opt) I 'Cald

LEA (EIN =1 => 4 [0] => [@+ ] = 1 7 1 ka -7 k HE (b) 2 f(x,y) sk ; x²4 y ² 282 x² + y2 >82. 2 let x = loso ya = 1 sino. } 2² = x² + y2 Here T x² + y²282 {(x,y) ER² 7.8,2 aga 2 => & cr. or <r. and 0 2 and -212 50 m ::$ = § 6,0) : 037, 48, -73042 Istemix) dx = ! 79 MAN gires, TI Ker drida w -

[k k ri de. - 1 อ 2 - => [kx2 ] = (SVN SVN => k. X2 [ 1 / 2 + - ） 2 7 kara = 2 7 k 2 152 in both the case after getting k we Can $62,8) > 0 see conditions both the Hence Satisfied are