Page 66 - Probability and Statistical Inference

1. Notions of Probability 43

The exponential Distribution: This is a very special distribution which
coincides with the Gamma(1,β) distribution where β(> 0) is referred to as a
parameter. This distribution is widely used in reliability and survival analyses.
The associated pdf is given by

where 0 < β < ∞. This distribution is also skewed to the right.

Figure 1.7.5 Exponential Densities: (a) β = 1 (b) β = 2

The pdf given by (1.7.23) has been plotted in the Figure 1.7.5 with β = 1, 2.
The plot (a) corresponds to the pdf of the standard exponential variable. The
plot (b) corresponds to β = 2 and it is clear that this pdf has moved to the rhs
compared with the standard exponential pdf.

As the parameter β increases, the pdf given by (1.7.23) will move farther to
the right and as a result the concentration in the right hand vicinity of zero will
decrease. In other words, as β increases, the probability of failure of items in
the early part of a life test would decrease if X represents the life expectancy of
the tested items.
The reader can easily check that the corresponding df is given by

In the context of a reliability experiment, suppose that X represents the
length of life of a particular component in a machine and that X has the
exponential distribution defined by (1.7.23). With any two fixed positive