Question

The Answer for i is suppose to be 0.625, but I am having trouble getting that number

Exercise 5.5. Consider the sample space S = {(x,y) € R2 : 22 + y2 <1}, with event space & suitably chosen, and with probability measure P determined by Area(E) Area(E) P(E) = Area(S) TT for E E E. Let X: S+R be the random variable defined by -1/2 if x < 0 and y 0, 1/3 if x < 0 and y < 0, Xx, y)) = 2 if x > 0 and y < 0, V x2 + y2 if x > 0 and y > 0. 26 Find the probability mass and density functions associated to X and use them to compute the following statistics. (i) E[X] (ii) Var[X] (iii) E[X3] (iv) E[(2X +3)31

Answer 1

solution;

given that

the sample space equation

S = {(x,y) $\epsilon$ R^2: $^{x2}$ +$^{y2}$$\leq$1}

the probability of P(E)=area(E)/area (S)

=area(E)/$\pi$

Solution - Given that B C D * To be Eoch hacoing Equal Orea = T(0) 4 T 4 P(B) = P(220.420) 1/4 Alen of s 77 14 P (B) = 14 Similarly, PC) - P(x20.4c0) 14/7 Pixzos YcO] <1/14/12 Pfxz0..420] -1741 244 - 1/4 P (o] P(A)

* The probability function is given by X(214) -1/2 13 2 P(x) 14 14 i €(x) En f(a) = *.* 144 +10) 174) +27944) Jolayn 200 yo O = 1/2 Y=1 dh dy into poby 28 COS y = 8 sino atyrore THA 14120 €(x) --118 +/12 + 1/2 2 +/ - dido 4 O' T20

Y2 -3+212 ор D T X X 1/2 X 14 24 Q 1/4 + 1 31 +27 = 0-65 €) = ( 18 14 ( % van * to's van de ++ tv +1+1} /0743 Jolyo di do 0720 9+4+144 9x4x4 + 1/16 do 157 144 + 1.188 Cir)

(c + -48*74 + 1/273 74 + BxY4 + 16144) นาง 13h Sandy so Do 9 = { set ya * * il fondo - 94 6-4ed =12 Thi g3 g do dol do do 946-4px 4px + 6 +158)_] 720 2.03 (d) € 3 ] € (421+33) € (823 + 24 + 33+ 3(on. € (82% + 24 +3624 +5341) SER +27 +36 € (+54 EC%)

=8X (205) +27 + 36 x (1.186)+54 X10:65) 121.268 V(x) € 02 - [ECX))? = 1.188-1065)? 0.7655