Answer to Question #289549 in Statistics and Probability for lusientoo

Question

A box contains 10 balls. Three are numbered 0, two are numbered 1, one is numbered 2, two are numbered 3, and two are numbered 4. The balls are mixed and one is selected at random. After a ball is selected, its number is recorded. Then it is replaced. If the experiment is repeated many times, find the standard deviation of the numbers on the balls. (Round off your final answer to 2 decimal places).

Answer 1

Let X be a random variable represents the number on the ball, then

$P(X=0)=0.3$

$P(X=1)=0.2$

$P(X=2)=0.1$

$P(X=3)=0.2$

$P(X=4)=0.2$

$\sigma=\sqrt{D}=\sqrt{M(X^2)-M^2(X)}$

$M(X)=0*0.3+1*0.2+2*0.1+3*0.2+4*0.2=1.8\implies M^2(X)=3.24$

$M(X^2)=0^2*0.3+1^2*0.2+2^2*0.1+3^2*0.2+4^2*0.2=5.6$

So, $\sigma=\sqrt{5.6-3.24}=\sqrt{2.36}\approx1.54$