Answer:
a) [tex]E(X) = 6.45[/tex]
b) [tex]E(X^{2} )= 57.25[/tex]
c) [tex]V(X) = 15.648[/tex]
d) E(3X + 2) = 21.35
e) [tex]E(3X^{2} +2) = 173.75[/tex]
f) V(3X+2) = 140.832
g) E(X+1) = 7.45
h) V(X+1) = 15.648
Step-by-step explanation:
a) [tex]E(X) = \sum xP(x)[/tex]
[tex]E(X) = (1*0.05) + (2*0.10) + (4*0.35) + (8*0.40) + (16*0.10)\\E(X) = 6.45[/tex]
b)
[tex]E(X^{2} ) = (1^{2} *0.05) + (2^{2} *0.10) + (4^{2} *0.35) + (8^{2} *0.40) + (16^{2} *0.10)\\ E(X^{2} )= 57.25[/tex]
c)
[tex]V(X) = E(X^{2} ) - (E(X))^{2} \\V(X) = 57.25 - 6.45^{2} \\V(X) = 15.648[/tex]
d)
[tex]E(3X+2) = 3E(X) + 2\\E(3X+2) = (3*6.45) + 2 \\E(3X+2) = 21.35[/tex]
e)
[tex]E(3X^{2} +2) = 3E(X^{2} ) + 2\\E(3X^{2} +2) = (3*57.25) + 2 \\E(3X^{2} +2) = 173.75[/tex]
f)
[tex]V(3X+2) = 3^{2} V(X)\\V(3X+2) = 9*15.648\\V(3X+2) = 140.832[/tex]
g)
[tex]E(X+1) = E(X) + 1\\E(X+1) = 6.45 + 1\\E(X+1) =7.45[/tex]
h)
[tex]V(X+1) = 1^{2} V(X)\\V(X+1) = 15.648[/tex]
