The exponential distribution as a waiting-time model
The exponential distribution is one of the most important continuous probability models because it describes
the waiting time until an event happens when events occur at a constant average rate. If \(X\sim \mathrm{Exp}(\lambda)\),
the parameter \(\lambda>0\) is the rate. Larger \(\lambda\) means shorter typical waiting times,
while smaller \(\lambda\) means longer waiting times.
PDF, CDF, and survival function
The probability density function (PDF) of the exponential distribution is
\[
f(x)=\lambda e^{-\lambda x}\quad (x\ge 0),
\]
and \(f(x)=0\) for \(x<0\). The PDF is “highest” at \(x=0\) and then decays smoothly, which matches the idea that
a long wait becomes less likely as time increases. Integrating the PDF gives the cumulative distribution function (CDF):
\[
F(x)=P(X\le x)=1-e^{-\lambda x}\quad (x\ge 0).
\]
Closely related is the survival function, the probability that the wait exceeds \(x\):
\[
P(X>x)=1-F(x)=e^{-\lambda x}.
\]
This is often the easiest way to compute tail probabilities. For example, with \(\lambda=0.5\),
\(P(X>2)=e^{-0.5\cdot 2}=e^{-1}\approx 0.3679\).
Mean, variance, and interpretation
Two summary quantities are especially useful:
\[
\mathbb{E}[X]=\frac{1}{\lambda}, \qquad \mathrm{Var}(X)=\frac{1}{\lambda^2}.
\]
So if \(\lambda=0.5\), the mean waiting time is \(1/0.5=2\) time units. In reliability and decay contexts,
\(\lambda\) can be interpreted as a “failure rate” or “decay rate.” The tool shows these values automatically so you can
connect the parameter to a concrete time scale.
Memoryless property (advanced note)
The exponential distribution has the memoryless property:
\[
P(X>s+t\mid X>s)=P(X>t).
\]
In words, if you have already waited \(s\) time units and the event still has not happened, your additional waiting time
has the same distribution as if you were starting from zero. This is one reason the exponential model is natural for
constant-rate processes, and it is also a key bridge to the Poisson process in more advanced probability.
How to use this tool
Enter a rate \(\lambda\) and choose what you want to compute: a survival probability \(P(X>x)\), a CDF value \(P(X\le x)\),
a PDF value \(f(x)\), or an interval probability \(P(a<X<b)\). The calculator outputs a clear step-by-step substitution,
along with mean and variance. The interactive plot shows the PDF and CDF together; the shaded region updates to match your
selected probability, and pressing Play moves a marker along the curves to help you visualize how the values change with \(x\).
