Answer to Question #313994 in Civil and Environmental Engineering for james

Question

An employee is selected from a staff of 10 to supervise a certain project by selecting a tag at random from a box containing 10 tags numbered from 1 to 10. (a) Find the formula for the probability distributions of X representing the number on the tag that is drawn. (b) What is the probability that the number drawn is less than 4? (c) Find the mean and variance of the random variable X.

Answer 1

An employee is selected from a staff of 10 to supervise a certain project by selecting a tag at random from a box containing 10 tags numbered from 1 to 10. Find the formula for the probability distribution of X representing the number on the tag that is drawn.

Find the mean and variance of the random variable X

Let random variable X represent the number of the drawn tag. Since the simple space consists of 10 equally likely events

$\Omega=\{1, 2, 3, 4 ,5 ,6 ,7 ,8 ,9 ,10\}$

thus the formula for probability distribution 

$f(x)=\begin{cases} \frac{1}{10} \ for \ x=1, 2, ...,10\\ 0 \ elsewhere \end{cases}$

The mean:

$\mu=\sum x_i P(x_i)=0.1\cdot(1+2+3+4+5+6+7+8+9+10)=5.5$

Variance:

$var(x)=\sum(x_i-\mu)^2P(x_i)=0.1\cdot((1-5.5)^2+(2-5.5)^2+(3-5.5)^2+(4-5.5)^2+(5-5.5)^2+(6-5.5)^2+(7-5.5)^2+(8-5.5)^2+(9-5.5)^2+(10-5.5)^2)=0.1\cdot(20.25+12.25+6.25+2.25+0.25+0.25+2.25+6.25+12.25+20.25)=8.25$