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 Standard Deviation (Y )

Answer 1

Since Each face is likely to occur

So probability of getting each face $=\dfrac{1}{8}$

The Probability Distribution for the 8-faces die is-

Mean is given by-

$E(Y)=\sum x_ip_i=(1\times\dfrac{1}{8})+(2\times\dfrac{1}{8})+(3\times\dfrac{1}{8})+(4\times\dfrac{1}{8})+(5\times\dfrac{1}{8})+(6\times\dfrac{1}{8})+(7\times\dfrac{1}{8})+(8\times\dfrac{1}{8})$

=$\dfrac{1+2+3+4+5+6+7+8}{8}=\dfrac{36}{8}=4.5$

Varaince is calculated by the formula-

$V(Y)=\dfrac{\sum(x_i-\bar{x})^2}{N}$

=$\dfrac{(3.5)^2+(2.5)^2+(1.5)^2+(0.5)^2 +(-0.5)^2+(-1.5)^2+(-2.5)^2+(-3.5)^2}{8}$

=$\dfrac{12.25+6.25+2.25+0.25+0.25+2.25+6.25+12.25}{8}$

=$\dfrac{42}{8}=5.25$

Standard deviation is given by-

$S(Y)=\sqrt{V(Y)}=\sqrt{5.25}=2.29$