Respuesta :
Answer:
a) 0.3012 = 30.12% probability that one of these light bulbs will last at least 24,000 hours.
b) 0.1813 = 18.13% probability that one of these light bulbs will last no longer than 4,000 hours.
c) 0.5175 = 51.75% probability that one of these light bulbs will last between 4,000 hours and 24,000 hours.
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]
Mean of 20,000 hours.
This means that [tex]m = 20000, \mu = \frac{1}{20000} = 0.00005[/tex]
a. At least 24,000 hours.
[tex]P(X > 24000) = e^{-0.00005*24000} = 0.3012[/tex]
0.3012 = 30.12% probability that one of these light bulbs will last at least 24,000 hours.
b. No longer than 4,000 hours.
[tex]P(X \leq 4000) = 1 - e^{-0.00005*4000} = 0.1813[/tex]
0.1813 = 18.13% probability that one of these light bulbs will last no longer than 4,000 hours.
c. Between 4,000 hours and 24,000 hours.
Less than 4000 or more than 24000:
We found in a and b, so
0.3012 + 0.1813 = 0.4825
Between these bounds:
1 - 0.4825 = 0.5175
0.5175 = 51.75% probability that one of these light bulbs will last between 4,000 hours and 24,000 hours.
The probability of one of these light bulbs will last at least 24,000 hours, no longer 4,000 hours, and between 4,000 hours and 24,000 hours is 23.12%, 18.13%, and 51.75% respectively.
What is probability?
Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1. Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.
Lucky Lumen light bulbs have an expected life that is exponentially distributed with a mean of 20,000 hours.
The exponential probability distribution, with mean m, will be
[tex]f(x) =\rm \mu e^{\mu x}\\\\Where \ \mu = \dfrac{1}{m}[/tex]
Then the probability that x is lower o equal to a is given by
[tex]P(X\leq x) = \int_0^a f(x) \rm \ dx[/tex]
Then we have
[tex]P(X\leq x) = 1 - e^{-\mu x}\\\\P(X > x ) = 1 - P(X\leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}[/tex]
Mean of 20,000 hours, then
[tex]\mu = \dfrac{1}{20,000}\\\\\mu = 0.00005[/tex]
a. At least 24,000 hours.
[tex]P(X > 24,000) = e ^{-0.00005 \times 24,000} = 0.3012 \ or \ 32.12\%[/tex]
The probability of one of these light bulbs will last at least 24,000 hours is 32.12%.
b. No longer than 4,000 hours.
[tex]P(X \leq 4,000) = e ^{-0.00005 \times 4,000} = 0.1813\ or \ 18.13\%[/tex]
The probability of one of these light bulbs will last no longer than 4,000 hours is 18.13%.
c. Between 4,000 hours and 24,000 hours.
Then we have a and b, so
0.3012 + 0.1813 = 0.4825
Between these bound
1 - 0.4825 = 0.5175
The probability of one of these light bulbs will last between 4,000 hours and 24,000 hours is 51.75%.
More about the probability link is given below.
https://brainly.com/question/795909
