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Wize University Statistics Textbook > Continuous Probability Distributions

Exponential Distribution

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Exponential Experiment


The exponential distribution describes the amount of time it takes until some event occurs. It is often used to model "failure" times or waiting times.

Examples
1) The length of time (in hours) between students dropping in at office hours.
2) The waiting time (in minutes) until a customer service operator answers your call.
3) The distance (in km) travelled by an old vehicle before it breaks down.

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An exponential experiment has the following properties:
  1. Memoryless property →\rightarrow this means the future probabilities are not influenced by how much time has elapsed in the past.
  • The distribution of the remaining life does not depend on how long the time as elapsed.
  • Example Although you've been on hold for a long time, it does not make it more likely that a customer service operator will answer your call in the next minute, or the minute after that.
  1. The exponential distribution is related to the Poisson distribution.
  • Recall in Poisson distribution that λ\lambda is the given success rate for a period of time. Given λ\lambda, we can determine the probability of something not occurring over a period of tt.

λ=\lambda= mean number of successes or occurrences in a given period of time

β=\beta= mean amount of time per success =1λ=\frac{1}{\lambda}


Example
I hiccup 4 times a minute.
  • λ=4\lambda=4 "successes" per minute
  • β=\beta= mean amount of time per hiccup ("success") =14=\frac{1}{4} (Translation: I hiccup every 1/4th minute or 15 seconds).

Watch Out!
"Success" just denotes the outcome that you are interested in.
"Failure" just denotes the opposite outcome (i.e. complement).

For example, these could be "male or female", "win or lose", "greater than 1 or not greater than 1", etc.

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Exponential Distribution

Recall

λ=\lambda= mean number of successes or occurrences in a given period of time
β=\beta= mean amount of time per success =1λ=\frac{1}{\lambda}

Let t=t= how much time has elapsed without something occurring.

Cumulative Distribution Function (CDF)



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Example #1
The number of burps follows an exponential probability distribution with an average of 3 burps a minute.

(a) What is the probability that the time between 2 burps is less than 2 minutes?

λ=3\lambda=3 "successes" per minute
β=\beta= mean amount of time per burp ("success") =1λ=13=\frac{1}{\lambda}=\frac{1}{3} (Translation: I burp every 1/3rd minute or 20 seconds).

P(X<t)=1−e−t/β=1−e−λtP(X<t)=1-e^{-t/\beta}=1-e^{-\lambda t}

P(X<2)=1−e−2/(13)=1−e−3(2)=0.9975P\left(X<2\right)=1-e^{-2/\left(\frac{1}{3}\right)}=1-e^{-3\left(2\right)}=0.9975


The high probability makes sense. Since I burp 3 times a minute (or I burp every 20 seconds), it is not very probable to have no burps a 2-minute gap between 2 burps. In other words, it is very probable that the time between 2 burps is less than 2 minutes.
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(b) What is the probability that I do not burp in the next two minutes?

This is the same way as asking: "What is the probability that it will take me longer than 2 minutes to burp?"


P(X>2)=e−3(2)=0.0025P\left(X>2\right)=e^{-3\left(2\right)}=0.0025

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Example #2
The time between a #23 bus arriving follows an exponential probability distribution with an average of 3 minutes. What is the probability that the time between 2 buses arriving is less than 2 minutes?

β=\beta= mean amount of time for bus to arrive=3=3 (Translation: A bus arrives an average of 3 minutes).

P(X<2)=1−e−t/β=1−e−2/3=0.4866P(X<2)=1-e^{-t/\beta}=1-e^{-2/3}=0.4866



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Do you see the differences between the two examples?

1. The first example tells you how many times a burp happens a minute.
2. The second example tells you the amount of time (in minutes) a bus arrives.

Watch Out!
Make sure you use λ\lambda and β\beta correctly!


The number of days that Bender the Robot can last with a fully charged battery can be modelled by an exponential distribution. He is expected to last β=100\beta=100 days before needing to recharge.

a) What is the probability that he will last longer than 120 days?

b) What is the probability that he will last fewer than 150 days?

c) Bender already lasted 120 days so far since his large full charge. "Eh.. I got another month," he said. Given that he lasted 120 days, what is the probability that he will last 30 more days (i.e. 150 days in total)?
Extra Practice
Exponential Distribution
The number of days that Bender the Robot can last with a fully charged battery can be modelled by an exponential distribution. He is expected to last β=100\beta=100 days before needing to recharge.
a) What is the probability that he will last longer than 120 days?
b) What is the probability that he will last fewer than 150 days?
Exponential Distribution
The number of days that Bender the Robot can last with a fully charged battery can be modelled by an exponential distribution. He is expected to last β=100\beta=100 days before needing to recharge.
a) What is the probability that he will last longer than 120 days?
b) What is the probability that he will last fewer than 150 days?
Exponential Distribution
On average, 2 apples fall from a tree every 45 seconds. Assuming it follows an exponential distribution, what is the probability that it will take between 30 seconds to a minute for 1 apple to drop?
Exponential Distribution
Pho Mo is a comedian who can make the audience laugh 4 times a minute. Suppose the time it takes him to make people laugh follows an exponential distribution. What is the probably that he gets no laughter in the first 30 seconds?
Exponential Distribution
The time (in hours) for Ming to repair a watch follows an exponential distribution with an average of 2 hours. A customer drops off a broken watch and tells Ming that he needs it in 2 hours. What is the probability that Ming will take longer than 2 hours to repair it?
Statistical Properties: Practice
True or false? Enter "T" for true; "F" for false (without quotations).
Exponential Distribution
Farid owns a jewelry shop in a bad side of town where he gets robbed an average of 2 times a week. A week has gone by but there has been no robbery. Which of the following is true?