Let's go through each part of the given question step-by-step.
c. Consider the required conditions for a discrete probability function, shown below.
[tex]$
\begin{array}{l}
f(x) \geq 0 \\
\Sigma f(x)=1
\end{array}
$[/tex]
To satisfy these conditions:
1. [tex]\( f(x) \geq 0 \)[/tex]: All probability function values must be greater than or equal to 0.
2. [tex]\( \Sigma f(x)=1 \)[/tex]: The sum of all probability function values must equal 1.
Given the discrete probability distribution:
[tex]\[
\begin{array}{l}
1: 0.2 \\
2: 0.3 \\
3: 0.1 \\
4: 0.4
\end{array}
\][/tex]
Does this probability distribution satisfy equation (5.1)?
Yes, all the probability function values (0.2, 0.3, 0.1, 0.4) are greater than or equal to 0.
Does this probability distribution satisfy equation (5.2)?
Yes, the sum of all probability function values is [tex]\(0.2 + 0.3 + 0.1 + 0.4 = 1\)[/tex], which equals 1.
d. What is the probability a service call will take 2.7 hours?
The given probability distribution does not explicitly include an entry for 2.7 hours. Since 2.7 hours is not a specified value in the provided probability distribution, the probability of a service call taking exactly 2.7 hours is 0.
e. A service call has just come in, but the type of malfunction is unknown. It is 3:00 PM and service technicians usually get off at 5:00 PM. What is the probability the service technician will have to work overtime to fix the machine today?
Overtime is necessary if the service call takes more than 2 hours because the service technicians get off at 5:00 PM and the call came in at 3:00 PM.
We need to calculate the probability that the service call will take more than 2 hours. This includes the events where the service call takes 3 or 4 hours.
Given the probabilities:
[tex]\[
\begin{array}{c}
P(X > 2) = P(3) + P(4) \\
P(X > 2) = 0.1 + 0.4 = 0.5
\end{array}
\][/tex]
So, the probability that the service technician will have to work overtime to fix the machine today is 0.5.
Summarizing:
d. The probability a service call will take 2.7 hours is [tex]\( \boxed{0} \)[/tex].
e. The probability the service technician will have to work overtime is [tex]\( \boxed{0.5} \)[/tex].
