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Exponential Distribution
Introduction
An electrical engineer who is in charge of an electrical wiring in a premise wants to know the number of faults in a given length of wire and also the distance between such faults.
He can analyzed the number of faults using the Poisson distribution. The number of faults along the wire maybe shown to give rise to the exponential distribution as defined below:
Definition
The general formula for the probability density function of the exponential distribution is is a parameter which is a given constant
. The following is the plot of the exponential probability density function.
If then the cumulative distribution function of the exponential distribution is
The following is the plot of the exponential cumulative distribution function.
Mean and variance of an exponential function
The mean and variance of an exponential function are respectively
Example 28
If jobs arrive every 15 seconds on average, λ= 4 per minute, what is the probability of waiting less than or equal to 30 seconds, i.e .5 min
Example 29
If , find , and
Example 30
Refer to Example 26. The random variable has a pdf which is an exponential distribution.
The Poisson Process
Recall that the Poisson distribution is often used to model the number o f events that occur in a given region of time or space. We say that the events follow a Poisson process with rate parameter when the number of events in disjoint intervals of time or space are independent and the number of events that occur in any interval of length has a Poisson distribution with mean . In other words when
The exponential distribution is the correct model for waiting times of an event whenever an event follows a Poisson process.
Applications of the Exponential Distribution
1. Time between telephone calls
2. Time between machine breakdowns
3. Time between successive job arrivals at a computing centre
Example 31
A radioactive mass emits particles according to a Poisson process at a mean rate of 15 particles per minute. At some point, a clock is started. What is the probability that more than 5 seconds will elapse before the next particle is emitted?
Example 32
Average rate of arrival of a person in a queue is 1.5 per minute. Determine the probability that (a) at most four person will arrive in any given minute
(b) at least five will arrive during an interval of 2 minutes
(c) at most 20 will arrive during an interval of 6 minutes
Lack of memory property/ Memoryless or Markov property
Scenario:
Suppose when a person arrives, one computer workstation has just been occupied (engaged) while another workstation has been occupied since (say 50 minutes) long. Then the probability distribution of the length of waiting time (to use the workstation) will be the same for either workstation. Therefore, it does not matter which workstation the person decides to wait!
The exponential distribution is memoryless. This means that the past does not influence the future. In mathematical terms, this means
The exponential distribution is memoryless .This means that the past does not influence the future. In mathematical terms, this means
For any
If and and are positive numbers then,
Example 33
The lifetime of a transistor in a particular circuit has an exponential distribution with mean 1.25 years. Find the probability that the circuit last longer than two years.
Example34
Refer to example 33. Assume the transistor is now three years old and is still functioning. Find the probability that it function for more than two additional years. Compare your answer with Example 33
Exercise
The length of time for one person to be served at a cafeteria is a random variable having an exponential distribution with a mean of 4 minutes. Find the probability that a person is served in less than 3 minutes on at least 4 of the next 6 days.
Introduction
An electrical engineer who is in charge of an electrical wiring in a premise wants to know the number of faults in a given length of wire and also the distance between such faults.
He can analyzed the number of faults using the Poisson distribution. The number of faults along the wire maybe shown to give rise to the exponential distribution as defined below:
Definition
The general formula for the probability density function of the exponential distribution is is a parameter which is a given constant
. The following is the plot of the exponential probability density function.
If then the cumulative distribution function of the exponential distribution is
The following is the plot of the exponential cumulative distribution function.
Mean and variance of an exponential function
The mean and variance of an exponential function are respectively
Example 28
If jobs arrive every 15 seconds on average, λ= 4 per minute, what is the probability of waiting less than or equal to 30 seconds, i.e .5 min
Example 29
If , find , and
Example 30
Refer to Example 26. The random variable has a pdf which is an exponential distribution.
The Poisson Process
Recall that the Poisson distribution is often used to model the number o f events that occur in a given region of time or space. We say that the events follow a Poisson process with rate parameter when the number of events in disjoint intervals of time or space are independent and the number of events that occur in any interval of length has a Poisson distribution with mean . In other words when
The exponential distribution is the correct model for waiting times of an event whenever an event follows a Poisson process.
Applications of the Exponential Distribution
1. Time between telephone calls
2. Time between machine breakdowns
3. Time between successive job arrivals at a computing centre
Example 31
A radioactive mass emits particles according to a Poisson process at a mean rate of 15 particles per minute. At some point, a clock is started. What is the probability that more than 5 seconds will elapse before the next particle is emitted?
Example 32
Average rate of arrival of a person in a queue is 1.5 per minute. Determine the probability that (a) at most four person will arrive in any given minute
(b) at least five will arrive during an interval of 2 minutes
(c) at most 20 will arrive during an interval of 6 minutes
Lack of memory property/ Memoryless or Markov property
Scenario:
Suppose when a person arrives, one computer workstation has just been occupied (engaged) while another workstation has been occupied since (say 50 minutes) long. Then the probability distribution of the length of waiting time (to use the workstation) will be the same for either workstation. Therefore, it does not matter which workstation the person decides to wait!
The exponential distribution is memoryless. This means that the past does not influence the future. In mathematical terms, this means
The exponential distribution is memoryless .This means that the past does not influence the future. In mathematical terms, this means
For any
If and and are positive numbers then,
Example 33
The lifetime of a transistor in a particular circuit has an exponential distribution with mean 1.25 years. Find the probability that the circuit last longer than two years.
Example34
Refer to example 33. Assume the transistor is now three years old and is still functioning. Find the probability that it function for more than two additional years. Compare your answer with Example 33
Exercise
The length of time for one person to be served at a cafeteria is a random variable having an exponential distribution with a mean of 4 minutes. Find the probability that a person is served in less than 3 minutes on at least 4 of the next 6 days.