Question

5. Let X represents the number of computers in an Australian household, for those that own a
computer.

a. Find and interpret the expected number of computers in a randomly selected Australian
household. (2 marks)
b. What is the probability that a randomly selected Australian household will have more than 2
computers?
c. Find V (4X +2). (1 mark)
d. Find E(30X + 20]. (1 mark)

Answer 1

$\begin{matrix} x & 1 & 2 & 3 & 4 & 5 \\ p(x) & 0.25 & 0.33 & 0.17 & 0.15 & 0.10 \end{matrix}$

a. The expected number of computers in a randomly selected Australian household

$$E(X)=1(0.25)+2(0.33)+3(0.17)+4(0.15)+5(0.10)=$$ $$=2.52$$

The expexted number of computers in an Australian household, for those that own a computer is

$2.52$ computers.

b. The probability that a randomly selected Australian household will have more than 2 computers is

$$P(X>2)=0.17+0.15+0.10=0.42$$

c.

$$V(X)=(1-2.52)^2(0.25)+(2-2.52)^2(0.33)+$$ $$+(3-2.52)^2(0.17)+(4-2.52)^2(0.15)+$$ $$+(5-2.52)^2(0.10)=1.6496$$

$$V(4X+2)=4^2\cdot V(X)=16\cdot1.6496=26.3936$$

d.

$$E(30X+20)=30E(X)+20=30\cdot2.52+20=95.6$$