Question

Recall the example of rolling a six-sided die. This is an example of a discrete uniform random variable, so named because the probability of observing each distinct outcome is the same, or uniform, for all outcomes. Let Y be the discrete uniform random variable that equals the face-value after a roll of an eight-sided die. (The die has eight faces, each with number 1 through 8.) Calculate E(Y ), Var(Y ), and StdDev(Y )

Answer 1

We have that sample space is $S=\{1,2,3,4,5,6,7,8\}$

The die is fair thus each of the eight faces has an equally likely probability of occurring, i.e., 1/8.

The expected value is calculated by the formula:

$E(Y) =\sum yP(y)$

$(1+2+3+4+5+6+7+8)\cdot\frac{1}{8}=4.5$

Variation is calculated by the formula:

$var(Y)=E(Y-E(Y)^2)=\sum(y-E(Y))^2P(y)=\sum(y-E(Y))^2\cdot\frac{1}{8}$

$((1-4.5)^2+(2-4.5)^2+(3-4.5)^2+(4-4.5)^2+(5-4.5)^2+(6-4.5)^2+(7-4.5)^2+(8-4.5)^2)\cdot\frac{1}{8}=5.25$

Standard deviation is calculated by the formula:

$stddev(Y)=\sqrt{var(Y)}=\sqrt{5.25}=2.29$

Answer:

E(Y) = 4.5

var(Y) = 5.25

steddev(Y) = 2.29