Question

Which of the following is the standard deviation of the random variable x, the number that appears on a die, if the probability of any number on a fair, 9-sided die is 1/9?
A.2.582
B.2.739
C.0.11
D.4.5

Answer 1

Answer with explanation:

When a dice with 9 faces are rolled total possible outcome =9

→x={1,2,3,4,5,6,7,8,9}

Probability of getting any number from 1 to 9 , when the die is rolled once

               $=\frac{1}{9}$

$E(x)=1\times \frac{1}{9}+2\times \frac{1}{9}+3\times \frac{1}{9}+4\times \frac{1}{9}+5\times \frac{1}{9}+6\times \frac{1}{9}+7\times \frac{1}{9}+8\times \frac{1}{9}+9\times \frac{1}{9}\\\\E(x)=(1+2+3+4+5+6+7+8+9)\times \frac{1}{9}\\\\E(x)=5$

$E(x^2)=1^2\times \frac{1}{9}+2^2\times \frac{1}{9}+3^2\times \frac{1}{9}+4^2\times \frac{1}{9}+5^2\times \frac{1}{9}+6^2\times \frac{1}{9}+7^2\times \frac{1}{9}+8^2\times \frac{1}{9}+9^2\times \frac{1}{9}\\\\E(x)=(1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2+9^2)\times \frac{1}{9}\\\\E(x)=\frac{9 \times 10 \times 19\times 1}{6 \times 9}\\\\E(x)=\frac{95}{3}$

Standard Deviation

     $=\sqrt{E(x^2)-(E(x))^2}\\\\=\sqrt{\frac{95}{3}-5^2}\\\\=\sqrt{\frac{95-75}{3}}\\\\=\sqrt{\frac{20}{3}}\\\\=\sqrt{6.67}\\\\=2.5819\\\\=2.582$  

Option A→2.582          

Answer 2

I got C as the answer