EX 3.7 number of computers in use

1. The Cal Poly Department of Statistics has a lab with six computers reserved for statistics majors.

   • Let $X$ denote the number of these computers that are in use at a particular time of day.
   • Probability distribution of $X$ is as follows:

   x	0	1	2	3	4	5	6
   p(x)	.05	.10	.15	.25	.20	.15	.10

2. Elementary probability properties are used to calculate other probabilities of interest.

• Example 1:

  • The probability that at most 2 computers are in use:
  $$\begin{array}{rl}P(X\le 2)& =P(X=0\text{ or }1\text{ or }2)\\ & =p(0)+p(1)+p(2)\\ & =.05+.10+.15=.30\end{array}$$

• Example 2:

  • The probability that at least 3 computers are in use:
    • Since the event “at least 3 computers are in use” is complementary to “at most 2 computers are in use”:
    • $\begin{array}{rl}P(X\ge 3)& =1-P(X\le 2)\\ & =1-.30=.70\end{array}$
    • Alternatively, this can also be obtained by adding together probabilities for $X=3,4,5,6$.

• Example 3:

  • The probability that between 2 and 5 computers inclusive are in use:
  • \begin{align}&P(2 \leq X \leq 5) \\ &= P(X = 2, 3, 4, \text{or } 5) \\&= .15 + .25 + .20 + .15 = .75\end{align}

• Example 4:

  • The probability that the number of computers in use is strictly between 2 and 5:
  • $P(2<X<5)=P(X=3\text{ or }4)=.25+.20=.45$