Cox(p. 22, 1966) defines the exponential distribution as follows:
Let X be the interval from the time origin to the first event. We can
obtain the the distribution of X from first principles or derive it from
(1). For no event occurs in (0,x] if and only if X > x. Hence
prob(X > x) = prob( N_x = 0) = e^{-\lambda x}
Thus F_X(x), the distribution function (d.f.) of X, and f_X(x), the
probability density function (p.d.f.), are
F_X(x) = 1 - e^{-\lambda x} (x \ge 0)
f_X(x) = F^'_X(x) = \lambda e^{-\lambda x} (x \ge 0)
We call this the exponential distribution of parameter \lambda.
Reference:
Cox, D.R.(1966) The Statistical Analysis of Series of Events. John Wiley
& Sons, NY.
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My question is, if this is the definition of an exponential distribution,
then what is the definition of an exponential stochastic process?
--V. Stokes
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