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.

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The PDF is \(f(x)= \displaystyle\frac{1}{10} e^{- \frac{1}{10} x}\)

\(\textbf{2)}\) What is \(\lambda\)?

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\(\lambda=\displaystyle\frac{1}{10}\)

\(\textbf{3)}\) State the cumulative density function.

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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?

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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?

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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?

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The answer is \(e^{- \frac{1}{10} (12)}\approx 0.30119\)

\(\textbf{7)}\) What is the mode?

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The mode is \( 0\)

\(\textbf{8)}\) What is the variance?

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The variance is \(100\)

\(\textbf{9)}\) What is the median?

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The median is \(10 \ln{2} \approx 6.9315\)

\(\textbf{10)}\) What is the mean?

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The mean is \(10\)

See Related Pages\(\)

\(\bullet\text{ Statistics Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Uniform Distribution}\)
\(\,\,\,\,\,\,\,\,p(x)=\frac{1}{b-a}…\)

\(\bullet\text{ Binomial Distribution}\)
\(\,\,\,\,\,\,\,\,p(r)={}_{n}C_{r}(p)^r(1-p)^{n-r}…\)

\(\bullet\text{ Poisson Distribution}\)
\(\,\,\,\,\,\,\,\,P(x)=\displaystyle\frac{\lambda^x e^{-\lambda}}{x!}…\)

\(\bullet\text{ Continuity Correction}\)
\(\,\,\,\,\,\,\,\,c-.5\lt x\lt c+.5…\)

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