91Ó°ÊÓ

In contrast to probability and possibility, which provide some clues to objective reality, plausibility is simply a subject-related concept: plausibility can only exist because it is borne by human reasoning. To put it another way, something is only conceivable if someone asserts it to be so.

Given: exponential distribution with \(\lambda = 1.\)

Given observations:

\(\begin{array}{l}82.8,11.6,359.5,502.5,307.8,179.7,242,26.5,244.8,304.3,379.1,212.6,\\229.9,558.9,366.7,204.6\end{array}\)

Sort the observations from smallest to largest:

\(\begin{array}{l}11.6,26.5,82.8,179.7,204.6,212.6,229.9,242,244.8,304.3,307.8,359.5,\\366.7,379.1,502.5,558.9\end{array}\)

We note that the data contains \(16\) data values.

We then need to determine the exponential -percentiles, which are the values of x for which the cumulative distribution function of the exponential distribution with \(\lambda = 1\)is equal to the probability

\(\frac{{i - 0.5}}{{16}}\) (or the closest probability). Note: These percentiles are the same as the Weibull percentiles (because the standard Weibull distribution is the exponential distribution with \(\lambda = 1)\)and thus the percentiles are \(\ln ( - \ln (1 - p))\)

\(i = 1 \Rightarrow p = \frac{{1 - 0.5}}{{16}} = 0.03125 \Rightarrow z = - 3.45\)

\(\begin{array}{l}i = 2 \Rightarrow p = \frac{{2 - 0.5}}{{16}} = 0.09375 \Rightarrow z = - 2.32\\i = 3 \Rightarrow p = \frac{{3 - 0.5}}{{16}} = 0.15625 \Rightarrow z = - 1.77\\i = 4 \Rightarrow p = \frac{{4 - 0.5}}{{16}} = 0.21875 \Rightarrow z = - 1.40\\i = 5 \Rightarrow p = \frac{{5 - 0.5}}{{16}} = 0.28125 \Rightarrow z = - 1.11\end{array}\)

\(\begin{array}{l}i = 6 \Rightarrow p = \frac{{6 - 0.5}}{{16}} = 0.34375 \Rightarrow z = - 0.86\\i = 7 \Rightarrow p = \frac{{7 - 0.5}}{{16}} = 0.40625 \Rightarrow z = - 0.65\\i = 8 \Rightarrow p = \frac{{8 - 0.5}}{{16}} = 0.46875 \Rightarrow z = - 0.45\\i = 9 \Rightarrow p = \frac{{9 - 0.5}}{{16}} = 0.53125 \Rightarrow z = 0.45\\i = 10 \Rightarrow p = \frac{{10 - 0.5}}{{16}} = 0.59375 \Rightarrow z = 0.65\\i = 12 \Rightarrow p = \frac{{12 - 0.5}}{{16}} = 0.71875 \Rightarrow z = 1.11\end{array}\)

\(\begin{array}{l}i = 13 \Rightarrow p = \frac{{13 - 0.5}}{{16}} = 0.78125 \Rightarrow z = 1.40\\i = 14 \Rightarrow p = \frac{{14 - 0.5}}{{16}} = 0.84375 \Rightarrow z = 1.77\\i = 15 \Rightarrow p = \frac{{15 - 0.5}}{{16}} = 0.90625 \Rightarrow z = 2.32\\i = 16 \Rightarrow p = \frac{{16 - 0.5}}{{16}} = 0.96875 \Rightarrow z = 3.45\end{array}\)

EXPONENTIAL PROBABILITY PLOT

To determine if the distribution of values is approximately exponentially distributed, we have to create an Exponential probability plot.

A normal probability plot is basically a scatterplot with the observations on the horizontal axis and the exponential percentiles on the vertical axis.

If the pattern in the exponential probability plot is roughly linear and does not contain strong curvature, then it is safe to assume that the distribution of the observations is approximately exponential.

The created exponential probability plot contains no strong curvature, thus the distribution of the observations could be approximately exponentially distributed.