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Sagot :
Answer:
0.1353 = 13.53% probability that the lifetime exceeds the mean time by more than 1 standard deviations
Step-by-step explanation:
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
[tex]f(x) = \mu e^{-\mu x}[/tex]
In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.
The probability that x is lower or equal to a is given by:
[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]
Which has the following solution:
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
The probability of finding a value higher than x is:
[tex]P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}[/tex]
The mean time for the component failure is 2500 hours.
This means that [tex]m = \frac{2500}, \mu = \frac{1}{2500} = 0.0004[/tex]
What is the probability that the lifetime exceeds the mean time by more than 1 standard deviations?
The standard deviation of the exponential distribution is the same as the mean, so this is P(X > 5000).
[tex]P(X > x) = e^{-0.0004*5000} = 0.1353[/tex]
0.1353 = 13.53% probability that the lifetime exceeds the mean time by more than 1 standard deviations
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