Exponential Decay Distribution

Exponential Decay Distribution#

The exponential decay distribution is a continuous distribution,

\[ f(t) = \lambda e^{-\lambda t}~, \qquad t \ge 0 \]

  • Continuous analogy to the geometric distribution.

  • \(\lambda\) : Interpret as the probability frequency of the event.

    You can see this setting \(x = 1/\lambda\) then \(P(X>x) = 1/e\). So \(1/\lambda\) is the period to decrease the probability that \(P(X>x)\) by a factor of \(e^{-1}\).

CDF : $\( F(t) = 1 - e^{-\lambda t}~ \)$

Survival Function : The compelement of the CDF is known as the survival function because it is the chance that the event occurs (usually some death) after some time \(t\). $\( P(T>t) = e^{-\lambda t} \)$

Expectation : By the tail sum formula, we can use the survival function for the expectation

$$
E(T) = \int_0^\infty P(X > t)~ dt = \int_0^\infty e^{-\lambda t}~ dt = \frac{1}{\lambda}
$$

Variance : Since the second moment is,

$$
E(T^2) = \frac{2}{\lambda}
$$

The variance is

$$
\text{Var}(T) = \frac{1}{\lambda^2}
$$

Median : The median of the exponential occurs when

$$
F(t) =  0.5
$$

Equivalently when the survival and cdf intersects

$$
P(T > t_{0.5}) = P(T \le t_{0.5})
$$

This allows us to simply solve $t$ when letting the survival function go to $50\%$,

$$
\begin{gather*}
e^{-\lambda t_{0.5}} = 0.5\\
\Big\Downarrow\\
t_{0.5} = \frac{\log(2)}{\lambda} = \log(2)E(T)
\end{gather*}
$$

Memoryless Property : Given an event has occur at \(T > t_1\), the chance that another event occur afterwards \(T > t_1 + t_2\) is independent of the first

$$
P(T > t_1 + t_2 \mid T > t_1) = P(T > t_2)
$$

::: Proof

$$
\begin{align*}
P(T > t_1 + t_2 \mid T > t_1) &= \frac{P(T > t_1 + t_2, T > t_1)}{P(T > t_t)} \\
&= \frac{P(T > t_1 + t_2)}{P(T > t_t)} \\
&= e^{-\lambda t_2}\\
&= P(T > t_2)
\end{align*}
$$
:::

Relationship to Poisson Distribution#

See Poisson Process.

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