The exponential distribution is a popular continuous probability distribution. It is usually used to model the elapsed time between events.
Notes
\({\text{Exponential Distribution}}\)
PDF
\( \lambda e^{- \lambda x} \)
CDF
\( 1- e^{- \lambda x} \)
Mean
\( \displaystyle\frac{1}{\lambda}\)
Median
\( \displaystyle\frac{\ln{2}}{\lambda} \)
Mode
\( 0\)
Variance
\( \displaystyle\frac{1}{\lambda^2} \)
Practice Problems
The amount of time people spend waking up each morning can be modeled by an exponential distribution with the average amount of time equal to ten minutes.
\(\textbf{1)}\) State the probability density function.
The PDF is \(f(x)= \displaystyle\frac{1}{10} e^{- \frac{1}{10} x}\)
\(\textbf{2)}\) What is \(\lambda\)?
\(\lambda=\displaystyle\frac{1}{10}\)
\(\textbf{3)}\) State the cumulative density function.
The CDF is \(1- e^{- \frac{1}{10} x}\)
\(\textbf{4)}\) What is the probabliity that a randomly selected person takes between 8 and 10 minutes to wake up?
The answer is \(e^{- \frac{1}{10} (8)}-e^{- \frac{1}{10} (10)}\approx 0.44933-0.36788 \approx 0.08145\)
\(\textbf{5)}\) What is the probabliity that a randomly selected person takes less than 7 minutes to wake up?
The answer is \(1- e^{- \frac{1}{10} (7)} \approx 0.50341\)
\(\textbf{6)}\) What is the probabliity that a randomly selected person takes more than 12 minutes to wake up?
The answer is \(e^{- \frac{1}{10} (12)}\approx 0.30119\)
