Download Exponential Distribution

Exponential Distribution
(Chapter 14)
M.I.G. McEachern High School
Exponential Distribution
Exponential distributions are directly related to
Poisson distributions
• If we have a discrete (a number we can count)
number of events within a specific time, then
there is a continuous (a number measured) time
interval between the events.
• Exponential distributions calculate the probability
of a certain time interval between events.
Exponential Distributions
• Thus, exponential distribution problems
are concerned with an event occurring
or not occurring with a specific time
interval.
• Related to Poisson Dist., if an event does not
occur, the number of events is zero, or x = 0.
Thus, as the Poisson Distribution would calculate:
𝜆0 𝑒 −𝜆
𝑃 0 =
= 𝑒 −𝜆
0!
• Moreover, if an event does not occur within the
specified interval, t, then the actual interval, T,
was larger. Or 𝑃 0 = 𝑃(𝑇 > 𝑡)
Exponential Distributions
• Stated again except using time intervals:
The probability that an event does not occur within
a specific time interval is:
𝑃 𝑇 > 𝑡′ =
−𝜆𝑡
𝑒
Where *T is the actual time (no numbers will be given)
*t is the proposed specific interval divided
by the time unit of lambda.
* is the average rate of occurrence.
Exponential Distributions
• Further, if an event does occur, this is the compliment of
an event not occurring. So 𝑃 𝑥 ≠ 0 = 1 − 𝑒 −𝜆
• Stated again except using time intervals would mean that
the actual time T would be less than the specified time, t.
The probability that an event occurs within a specific time
interval is:
𝑃 𝑇 < 𝑡′ = 1 −
−𝜆𝑡
𝑒
Where *T is the actual time (no numbers will be given)
*t is the proposed specific interval divided by the time unit of lambda.
* is the average rate of occurrence.
Exponential Distributiond
• Lastly, if a certain event occurs  times within
some interval u of time.
• Then the mean or average time between
events is:
𝑢
𝐸 𝑥 =
𝜆
Exponential Distribution
Example 1:
An electric motor's constant failure rate is 0.0004 failures/hr. Calculate
the probability of failure for a 150 hr mission.
 = 0.0004, t = 150 hr.
Further if the event is failure, and we want the probability this will
happen use:
𝑃 𝑇 < 𝑡 = 1 − 𝑒 −𝜆𝑡
So 𝑃 𝑇 < 150 hr. = 1 − 𝑒 −.0004(150) ≈ 0.0582
Exponential Distribution
Example 1:
If the electric motor's constant failure rate is 0.0004
failures/hr. Calculate probability of complete success for a 150 hr
mission.
 = 0.0004, t = 150 hr.
Further if the event is failure, and we want the probability this
does not happen use:
𝑃 𝑇 > 𝑡 = 𝑒 −𝜆𝑡
So 𝑃 𝑇 > 150 hr. = 𝑒 −.0004(150) ≈ 0.9418
Exponential Distribution
Example 1:
An electric motor's constant failure rate is 0.0004 failures/hr.
What is the Mean life expectancy E(x) = ?
The mean is found using 𝐸 𝑥 =
Thus, 𝐸 𝑥 =
1 hour
.0004
𝑢
𝜆
= 2,500 hours.
So on average, we see one failure every 2500 hours.
Exponential Distribution Practice
1. If two customers arrive every 30 seconds on average,
what is the probability of waiting less than or equal to 30
seconds for the next customer. What is the mean time
between customers?
𝑃 𝑇 < 30𝑠𝑒𝑐 = 1 − 𝑒 −2
𝐸 𝑇 =
1
= 0.865
𝑢 30 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
=
= 15 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
𝜆
2
2. Accidents occur with a Poisson distribution at an average
of 4 per week. Calculate the probability that at least two
days will elapse between accidents? What is the mean
time between accidents? 2
𝑃 𝑇 > 2 𝑑𝑎𝑦𝑠 =
−4 7
𝑒
𝑢
1𝑤𝑒𝑒𝑘
𝐸 𝑇 = =
= .25 𝑤𝑒𝑒𝑘𝑠
𝜆 4 𝑎𝑐𝑐𝑖𝑑𝑒𝑛𝑡𝑠
= 0.3189
𝑜𝑟
7 𝑑𝑎𝑦𝑠
= 1.75 𝑑𝑎𝑦𝑠
4 𝑎𝑐𝑐𝑖𝑑𝑒𝑛𝑡𝑠
T.O.D.
A CD player has an average record of successfully operating and
providing listening enjoyment for more than 5,000 hours on the
average before requiring repairs. A customer is planning on buying
a CD player for installation in a boat that will be taking an extended
cruise that will demand 4,000 hours of play before being able to
obtain repairs or routine maintenance. How reliable will the CD
player be for the customer? (Assume an exponential distribution)