Answer to Question #312774 in Statistics and Probability for jmark

Question

A box contains 8 balls. One is numbered 2, two are numbered 3, one is numbered 4, and four are numbered 5. 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 variance of the numbers on the balls. (Round off your final answer to 2 decimal places).

Answer 1

Let X - the random variable representing the number on the selected ball. To find the variance of the probability distribution, we can use the following formula:

$$\sigma^2=\sum(x_i-\mu)^2\cdot P(x_i)$$

where:

$x_i:$ the $i^{th}$ value

$\mu:$ the mean of the distribution

$P(x_i):$ the probability of the $i^{th}$ value

$\mu=2\cdot\cfrac{1}{8}+3\cdot\cfrac{2}{8}+4\cdot\cfrac{1}{8}+5\cdot\cfrac{4}{8}=4;\\$ $\\X-\mu=\begin{Bmatrix} 2-4, 3-4, 4-4, 5-4 \end{Bmatrix}=$

$=\begin{Bmatrix} -2, -1, 0, 1 \end{Bmatrix}$

$\sigma^2=(-2)^2\cdot \cfrac{1}{8}+(-1)^2\cdot \cfrac{2}{8}+0^2\cdot \cfrac{1}{8}+1^2\cdot \cfrac{4}{8}=$

$=\cfrac{10} {8} =1.25.$