II. Probability and Statistics for Reliability
A. Basic concepts
3. Discrete and continuous probability distributions (Analyze)
Compare and contrast various distributions (binomial, Poisson, exponential, Weibull, normal, log-normal, etc.) and their functions (e.g., cumulative distribution functions (CDFs), probability density functions (PDFs), hazard functions), and relate them to the bathtub curve.
This lesson takes a close look at the discrete distributions commonly used in reliability engineering.
Additional References
The Poisson Distribution (article)
Poisson Distribution Calculation (article)
Binomial Cumulative Density Function (article)
Binomial Probability Density Function (article)
Hypergeometric Distribution (article)
OC Curve with Hypergeometric Method (article)
OC Curve with Binomial Method (article)
Quick Quiz
1-19. An airline has a fleet of four-engine aircraft. Maintenance records indicate that, on average, an engine fails twice in 10,000 operating hours with normal preventive maintenance. Calculate the Poisson distributed probability that two or more engines on an aircraft will fail during a typical flight time of 7 hours.
(A) 0.001398
(B) 0.001399
(C) 0.0000009786
(D) 0.0000009791
(D) 0.0000009791
Use the Poisson as directed in the question and calculate the probability of exactly zero or one failure, then subtract the combined values from one to find the chance of 2 or more failures.
The expected number of failures, mu, over 7 hours given a failure rate of 2 per 10000 hours is ( 2 /10,000 ) * 7 = 0.0014.
P(0,0.0014) = (exp[-0.0014 * 0.0014^0) / 0! = 0.99860098
P(1,0.0014) = (exp[-0.0014 * 0.0014^1) / 1! = 0.00139902
For two or more we subtract the chance of zero and one from one (compliment) 1 – 0.99860098 – 0.00139902 = 0.0000009791
1-33 A manufacturing plant operates 9 units, at least 7 of which must be operating for production volume requirements to be met. If there is a .23 probability that a malfunction will occur for any particular unit, what is the probability that 7 units can remain operating throughout the day?
(A) 0.1605
(B) 0.1628
(C) 0.3510
(D) 0.4960
(C) 0.3510
My first thought was to use the k-out-of-n redundancy formula. I think it’s possible to use that formula if we convert the probability of failure, 0.23, to a reliability. The formula reduces to the binomial though and adds an extra step. So, let’s just use the binomial directly.
We need at least 7 units to operate of the 9. We can calculate the probability that none fail, x = 0, and the probability that exactly 1 unit will fail. Then add those together, which is the probability that 7 or more will operate for the day of production.
$$ P\left( 0,9,0.23 \right)=\left( \begin{array}{*{35}{l}}9 \\ 0 \\ \end{array} \right){{0.23}^{0}}{{\left( 1-0.23 \right)}^{9-0}}=0.0952$$
and for exactly one unit failing
$$ P\left( 1,9,0.23 \right)=\left( \begin{array}{*{35}{l}}9 \\1 \\\end{array} \right){{0.23}^{1}}{{\left( 1-0.23 \right)}^{9-1}}=0.2558$$
then add the two results to find the probability of at least 7 units operating for a day. 0.0952 + 0.2558 = 0.3510
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