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The [[marginal distribution]] (transient distribution) of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift ''μ '' and variance ''σ''<sup>2</sup> is
::<math>\mathbb P(Z(t) \leq z) = \Phi \left(\frac{z-\mu t}{\sigma t^{1/2}} \right) - e^{2 \mu z /\sigma^2} \Phi \left( \frac{-z-\mu t}{\sigma t^{1/2}} \right)</math>
for all ''t'' ≥ 0, (with Φ the [[cumulative distribution function of the normal distribution]]) which yields (for ''μ'' < 0) when taking t → ∞ an [[exponential distribution]]<ref name="harrison-book">{{cite book | title = Brownian Motion and Stochastic Flow Systems | first = J. Michael | last = Harrison | author-link = J. Michael Harrison | year = 1985 | publisher = John Wiley & Sons | isbn = 978-0471819394 | url =
::<math>\mathbb P(Z<z) = 1-e^{2\mu z/\sigma^2}.</math>
For fixed ''t'', the distribution of ''Z(t)'' coincides with the distribution of the running maximum ''M(t)'' of the Brownian motion,
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