Question

Cycle availability of a system. In the jargon of system maintenance, cycle availability is defined as the probability that the system is functioning at any point in time. The DoD developed a series of performance measures for assessing system cycle availability (Start, Vol. 11, 2004). Under certain assumptions about the failure time and maintenance time of a system, cycle availability is shown to be uniformly distributed between 0 and 1. Find the following parameters for cycle availability: mean, standard deviation, 10th percentile, lower quartile, and upper quartile. Interpret the results.

Short answer

The mean of a random variable X is 0.5.

The standard deviation is $\sigma =\frac{1}{\sqrt{12}}$.

The tenth percentile is 0.10.

The lower quartile is 0.25.

The upper quartile is 0.75.

Step-by-step solution

Given information

The failure and maintenance time of a system and cycle availability are uniformly distributed in intervals of 0 to 1.

Let X denotes the failure time and maintenance time of a system and cycle availability.

The probability density function of X is:

$f\left(x\right)=1;0⩽x⩽1$.

Finding the population parameters

The mean of a random variable X is:

$\begin{array}{rcl}\mu & =& \frac{1}{2}\\ & =& 0.5\end{array}$.

Since the mean of a uniform distribution is:$\mu =\frac{c+d}{2};c⩽x⩽d$.

The standard deviation is:

$\sigma =\frac{1}{\sqrt{12}}$.

For the uniform distribution, the standard deviation is:$\sigma =\frac{d-c}{\sqrt{12}};c⩽x⩽d$.

Also, $P\left(a<X<b\right)=\frac{b-a}{d-c};c⩽a<b⩽d$.

Therefore, the 10^th percentile is obtained as:

$\begin{array}{rcl}P\left(X⩽x\right)& =& 0.10\\ \frac{x-0}{1}& =& 0.10\\ x& =& 0.10\end{array}$.

Thus, the 10^th percentile is 0.10.

The lower quartile is obtained as:

$\begin{array}{rcl}P\left(X⩽x\right)& =& 0.25\\ \frac{x-0}{1}& =& 0.25\\ x& =& 0.25\end{array}$.

The lower quartile is 0.25.

The upper quartile is obtained as:

$\begin{array}{rcl}P\left(X⩽x\right)& =& 0.75\\ \frac{x-0}{1}& =& 0.75\\ x& =& 0.75\end{array}$.

The lower quartile is 0.75.

Interpretation of the population parameters

The mean of the random variable indicates that, on average, the probability that the system is functioning at any point in time is 0.5.

The standard deviation indicates the deviation of the probability from the average value is: $\frac{1}{\sqrt{12}}$.

The 10th percentile indicates that 10% of the probability of the system functioning at any point in time is 0.10.

The lower quartile indicates that 25% of the probability of the system functioning at any point is 0.25.

The lower quartile indicates that 75% of the probability of the system functioning at any point in time is 0.75.