OPMT 1197
Business Statistics
Lecture 11/12: Exponential and Uniform Distributions
Exponential Probability Distribution
Examples where it applies:
The length of time between telephone calls or between arrivals at a service station
The time required to load a truck
1. Used for continuous variables only since time is continuous.
2. Right skewed (mean > median) since the majority of the values (i.e. the waiting times) tend
to be on the left-side, or lower end, of the distribution.
3. The values can never be negative since waiting time cannot be negative.
It is often used to model the distribution of lifetimes for components, batteries and light bulbs.
It is used because of the “memoryless” property of the exponential distribution. If a component
has lasted 300 hours then the probability that it will last another 300 hours is the same as the
original distribution of lifetime. At each point the component has no effect of wear. The
component forgets that it already lasted 300 hours and starts fresh.
x
Cumulative Probability: PX x 1 e x and P X x e
• X = amount of time between arrivals • μ = the average time between arrivals (occurrences)
• Standard Deviation = Mean μ
1. The time it takes to load a truck at Schips loading dock averages 15 minutes. If the time it
takes to load a truck follows an exponential distribution, find the probability that it takes:
(a) 6 minutes or less to load a truck? (b) longer than 18 minutes to load a truck?
(c) between 6 and 18 minutes to load a truck? (d) Calculate the standard deviation.
The exponential distribution is closely related to the Poisson distribution. For example, the
Poisson distribution counts the number of customers that arrive at a bank during an hour while
the exponential distribution looks at the length of time between customers or the length of
time until the next customer arrives.
Poisson: Number of events over a time interval Exponential: Length of time between events
2. On Monday mornings, at a CIBC branch, an average of 12 customers arrive every hour.
The time between arrivals follows an exponential distribution.
(a) The average time between customers is?
(b) What is the probability that the next customer will arrive within the next 3 minutes?
(c) Given that no customers arrived on Monday morning between 10:15 and 10:25 am, what
is the probability that the next customer will arrive by 10:30 am?
Sol: 1. (a) 0.3297 (b) 0.3012 (c) 0.3691 (d) 15 minutes 2. (a) 5 minutes (b) 0.4512 (c) 0.6321
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Business Statistics
Lecture 11/12: Exponential and Uniform Distributions
Exponential Probability Distribution
Examples where it applies:
The length of time between telephone calls or between arrivals at a service station
The time required to load a truck
1. Used for continuous variables only since time is continuous.
2. Right skewed (mean > median) since the majority of the values (i.e. the waiting times) tend
to be on the left-side, or lower end, of the distribution.
3. The values can never be negative since waiting time cannot be negative.
It is often used to model the distribution of lifetimes for components, batteries and light bulbs.
It is used because of the “memoryless” property of the exponential distribution. If a component
has lasted 300 hours then the probability that it will last another 300 hours is the same as the
original distribution of lifetime. At each point the component has no effect of wear. The
component forgets that it already lasted 300 hours and starts fresh.
x
Cumulative Probability: PX x 1 e x and P X x e
• X = amount of time between arrivals • μ = the average time between arrivals (occurrences)
• Standard Deviation = Mean μ
1. The time it takes to load a truck at Schips loading dock averages 15 minutes. If the time it
takes to load a truck follows an exponential distribution, find the probability that it takes:
(a) 6 minutes or less to load a truck? (b) longer than 18 minutes to load a truck?
(c) between 6 and 18 minutes to load a truck? (d) Calculate the standard deviation.
The exponential distribution is closely related to the Poisson distribution. For example, the
Poisson distribution counts the number of customers that arrive at a bank during an hour while
the exponential distribution looks at the length of time between customers or the length of
time until the next customer arrives.
Poisson: Number of events over a time interval Exponential: Length of time between events
2. On Monday mornings, at a CIBC branch, an average of 12 customers arrive every hour.
The time between arrivals follows an exponential distribution.
(a) The average time between customers is?
(b) What is the probability that the next customer will arrive within the next 3 minutes?
(c) Given that no customers arrived on Monday morning between 10:15 and 10:25 am, what
is the probability that the next customer will arrive by 10:30 am?
Sol: 1. (a) 0.3297 (b) 0.3012 (c) 0.3691 (d) 15 minutes 2. (a) 5 minutes (b) 0.4512 (c) 0.6321
1 of 3
