Statistics 305A -- Handout
The Exponential Probability Plot
Equation (5.25) on page 258 gives the CDF of the Exponential E(α ) population as
F ( x) =
 1 − e − x / α

0

if x > 0
if x ≤ 0 .
The Standard Exponential distribution is E(1). If xp is the p-quantile in the population E(α ) and
yp is the corresponding p-quantile in the population E(1), then
y p = xp /α .
(1)
This is a straight line relationship having intercept zero and slope 1/α. A plot of ordered pairs
(xp, yp) of p-quantiles would be a plot of ordered pairs
( xp,
− ln (1 − p)
)
(2)
because for the E(1) distribution
F(yp ) = 1− e
−yp
= p
or
y p = − ln (1 − p) .
This plot of points (2) will be a line having intercept 0 and slope 1/α.
Now consider a population that is derived from E(α ) by adding the number t to each population
element. (Suppose t > 0). Quantiles in this derived population have value xp + t and the
scatterplot of points similar to (2) above, will be of ordered pairs
( x p + t,
)
− ln (1 − p) .
The plot will still be a line with slope 1/α but the line will intersect the horizontal axis at t not
zero. A random variable T defined on this derived population will not be an Exponential random
variable because it cannot assume any value in the interval (0, t). However, the random variable
S = T −t
will have the Exponential E(α ) distribution. Probabilities for T are computed as
1
P (T ≤ a) = P ( S + t ≤ a)
= P( S ≤ a − t )
= F (a − t ) .
The utility of these results is that whenever the ordered sample x1 ≤ x 2 ≤ K ≤ x n from a
population, paired with E(1) p-quantiles as

(i − 0.5)  

 xi , − ln  1 −
,
n  


i = 1, 2, K, n
(3)
gives a reasonably linear scatterplot, the evidence is that the sampled population has the shape of
an Exponential E(α ) population and 1/α is the slope of the linear trace. If the linear trace runs
approximately thru the origin, the sampled population is approximately exponential and the
random variable T defined on the sampled population has approximate distribution E(α ). If the
linear trace intersects the horizontal axis at t ≠ 0 then the random variable S = T – t is approximately E(α ).
Your textbook calls t the threshold value. What is given above is summarized in the first
paragraph on page 273. The scatterplot of ordered pairs (3) is called an Exponential Probability
Plot.
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