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Using the exponential distribution, it is found that:
a) There is a 0.0952 = 9.52% probability that a television set fails in less than 10,000 hours.
b) There is a 0.3012 = 30.12% probability that a television set lasts more than 120,000 hours.
c) There is a 0.1809 = 18.09% probability that a television set fails between 60,000 and 100,000 hours of use.
d) The 90th percentile is of 230,259 hours.
In probability theory and records, the exponential distribution is a non-stop possibility distribution that frequently issues the quantity of time till a few specific events happen. It is a method wherein activities show up continuously and independently at a regular common rate.
As an instance, the amount of time (starting now) until an earthquake occurs has an exponential distribution. Different examples consist of the duration, in mins, of long-distance enterprise phone calls, and the quantity of time, in months, a car battery lasts.
The lifetime of LCD TV sets follows an exponential distribution with a mean of 100,000 hours. Compute the probability a television set:
a. Fails in less than 10,000 hours.
b. Lasts more than 120,000 hours.
c. Fails between 60,000 and 100,000 hours of use.
d. Find the 90th percentile. So 10 percent of the TV sets last more than what length of time?
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Solution :
0.9
0.3012
0.1809
230258.
Given that:
μ = 100,000
λ = 1/μ = 1 / 100000 = 0.00001
a. Fails in less than 10,000 hours.
P(X < 10,000) = 1 - e^-λx
x = 10,000
P(X < 10,000) = 1 - e^-(0.00001 * 10000)
= 1 - e^-0.1
= 1 - 0.1
= 0.9
b. Lasts more than 120,000 hours.
X more than 120000
P(X > 120,000) = e^-λx
P(X > 120,000) = e^-(0.00001 * 120000)
P(X > 120,000) = e^-1.2
= 0.3011942 = 0.3012
c. Fails between 60,000 and 100,000 hours of use.
P(X < 60000) = 1 - e^-λx
x = 60000
P(X < 60,000) = 1 - e^-(0.00001 * 60000)
= 1 - e-^-0.6
= 1 - 0.5488116
= 0.4511883
P(X < 100000) = 1 - e^-λx
x = 100000
P(X < 60,000) = 1 - e^-(0.00001 * 100000)
= 1 - e^-1
= 1 - 0.3678794
= 0.6321205
Hence,
0.6321205 - 0.4511883 = 0.1809322
d. Find the 90th percentile. So 10 percent of the TV sets last more than what length of time?
P(x > x) = 10% = 0.1
P(x > x) = e^-λx
0.1 = e^-0.00001 * x
Take the In
−2.302585 = - 0.00001x
2.302585 / 0.00001
= 230258.5
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