ABSTRACT
The notation τ ∼ Exp(λ) means that Fτ (t) = P (τ ≤ t) = 1− e−λt, t > 0. (3.1.1)
The density function of r.v. τ is fτ (t) = λe−λt. The characteristic property of the exponential distribution is that the
so-called failure rate h(t) is constant:
h(t) = fτ (t)
1− Fτ (t) = λ = Const. (3.1.2)
The probabilistic meaning of (3.1.2) is the following. Suppose we know that a component whose lifetime τ ∼ Exp(λ) survived time t, i.e. τ > t. Then the conditional probability to fail in the interval [t, t + δt] does not depend on t:
P (τ ∈ [t, t+ δt]|τ > t) = e −λt − e−λ(t+δt)
e−λt = 1− e−λδt λ · δt.
