Question

Let X be a discrete random variable with the following PMF 6 for k € {-10,-9, -, -1,0, 1, ... , 9, 10} Px(k) = otherwise The random variable Y = g(X) is defined as Y = g(x) = {x if X < 0 if 0 < X <5 otherwise Calculate E[X], E[Y], var(X), and var(Y) for the two variables X and Y

Answer 1

X be a disorute random variable wohich haa +he 2ferke-0,-9,,09,10 +hense 10 E(x)kP CK) K1O 10 Σ. K Ke-io +-40f 9+-8 IN

Vase (x) ECx)- E2(x) E(x)-0 E)0] E(x2) 10 K2 (k) ECx2) K-1o Kz-10 k 2 21 K 0 & +9 102 22 21 2 (to 21 21 +8 to 10(t0+1) (a0 +) 24 n(n)(en 6 3 21 3 3

Var C) E(A)-Ex ECx)-0 Ecx) weget £(x2) 110 11 0 T : 3

andom variable y (x) io The Sin it 02x 25 0therneise 57 We see trat P(x0) PCY o) P(0KES) PCY x) (5) P(Y5) Then (x Lo) PCY 0 P (x=-10)P(x-+ f(x-8 Px0) timme + 21

P.b pyx)P(ox5) P(M)t Px)+ Px5), 4imes 21 Then) PCN x) 5 I usheu xt 5 PCY 5 P(x5 P(x6) + P(*T)+ +P(x10) 5 times 21 CrudiHonal Then, The bmf ofY T 57 it Lala R Lo

ENIX O.P(Y) rcy2) t55 + 21 * 5 TECYI] ELY) [a X+5 we got E(x) -0 5E(x) +5 5x 5 21 2 5 21 E(Yx) Py() ΣΗ 0,5

+ 25 x 21 21 E (v) EE() (x25 +25 33 3 185 x S 3 21 925 63 Vasr CY) E(Y) - () 125 = 13 .265 63 21 1 LalaLa AA 13