Question

Let X1 be a random variable whose value is the result of rolling an 8- sided die, and X2 a random variable whose value is the result of rolling a 12-sided die. (1) Find E(X1 + X2). (2) Find E(X{ + Xž). (3) Find V(X1 + X2).

Answer 1

Sof?: let Xe be a rondom variable whose value of the result of rolling on 8-sided die Probability moss function of X, is PIX = x ] = { # j < = 1,2,3,4,5, 5, 7,8 o ; → E (X,) = { p(x, = 34) 1+2+3+4+6 +7+8 +5 9 - → E (x,) 36 - 4.5 E (x2) = 5x2 P(X=) - 12+2? +32+4+5+62+72, 82 9 8(8+1) 8*2+1) 6 + 8 9*17 6 - E (43) - 51 .: V(x1) = E(X) - (E (X))2 51 2 . 81 4 = 102-81 4 = V(Xi) = 21 21

.: Xg is a random variable whose value is the result of rolling 12-sided olie. pmf of ty cp geven as a plxg = 42) = { ; 9=1,2, 3,...,12 زه ew K(12+1) = => E(X) = Enly PlXg = *) - 1+2+3+...+12 12 > Elxg) = 3 = 6.5 E 1X2) = EX2 P1X3 = x2) R+22+3? + olt 12 12 (1271) ( 1242 +1) 6*12 13 * 25 6 => E(X2) 3225 6 38 = 54.167 vorlxg) = E(X32) -(E (X))2 35 - (3) 325 169 2 = (5x)n5 6 4 143 12

(1) ECX, + x2) = E(X) + E1*2) + 13 6 11 22 = 11 2 =) E(X, +Xq) = 11 (2) E(x2+x2) = E(X2) +E(X2) 325 + 385 - is - 478 6 = 79.67 E14,2+xga) = 478 = 79. 67 (6x31+ lixon = (5%+421 ( 2 yuapuatapu? wes orp 414:.' = (5x+' 2 + 143 900- - 17.167 => 117,4%,0) 906 12 - 17.167