# 3 3) The time in hours during which an electrical generator is operational is a random variable that follows an exponential distribution. If the average operational time is 160 hours, what is the probability that a generator of this type will be operational between 60 and 160 hours? 4 Next > < Previous (DELL F3 E 4 DII F4 R 20 % 5 F5 T ·ő: < 6 F6 Y * & 7 U F8 * O prt sc F9 O 8 F10 home 0 0 F11 1/ P end
# 3 3) The time in hours during which an electrical generator is operational is a random variable that follows an exponential distribution. If the average operational time is 160 hours, what is the probability that a generator of this type will be operational between 60 and 160 hours? 4 Next > < Previous (DELL F3 E 4 DII F4 R 20 % 5 F5 T ·ő: < 6 F6 Y * & 7 U F8 * O prt sc F9 O 8 F10 home 0 0 F11 1/ P end
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Problem Statement:
#### Understanding the Probability of Generator Operational Time
The time in hours during which an electrical generator is operational is a random variable that follows an exponential distribution. If the average operational time is 160 hours, what is the probability that a generator of this type will be operational between 60 and 160 hours?
#### Detailed Explanation:
The problem involves an exponential distribution, which often models the time until a certain event occurs, such as the failure of a machine. In this case, it models the operational time of an electrical generator.
To solve this, you would typically use the properties of the exponential distribution and apply the relevant probability formulas. The probability density function of an exponential distribution is given by:
\[
f(x; \lambda) = \lambda e^{-\lambda x}
\]
where:
- \( \lambda \) (lambda) is the rate parameter, which is the reciprocal of the mean.
However, for practical exercises on an educational website, students would often be guided through:
1. **Identifying the Parameters**: Here, the average operational time is given as 160 hours, so the mean \( \mu = 160 \) hours. Given that \( \lambda = \frac{1}{\mu} \), therefore \( \lambda = \frac{1}{160} \).
2. **Probability Calculation**: The probability that the generator is operational between 60 and 160 hours can be calculated using the cumulative distribution function (CDF) of the exponential distribution:
\[
F(x; \lambda) = 1 - e^{-\lambda x}
\]
Therefore, you find the probability at 160 hours and subtract the probability at 60 hours.
For an educational website, it is important to follow through step-by-step calculations and perhaps include interactive tools or simulations for better understanding.
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Transcribed Image Text:### Problem Statement:
#### Understanding the Probability of Generator Operational Time
The time in hours during which an electrical generator is operational is a random variable that follows an exponential distribution. If the average operational time is 160 hours, what is the probability that a generator of this type will be operational between 60 and 160 hours?
#### Detailed Explanation:
The problem involves an exponential distribution, which often models the time until a certain event occurs, such as the failure of a machine. In this case, it models the operational time of an electrical generator.
To solve this, you would typically use the properties of the exponential distribution and apply the relevant probability formulas. The probability density function of an exponential distribution is given by:
\[
f(x; \lambda) = \lambda e^{-\lambda x}
\]
where:
- \( \lambda \) (lambda) is the rate parameter, which is the reciprocal of the mean.
However, for practical exercises on an educational website, students would often be guided through:
1. **Identifying the Parameters**: Here, the average operational time is given as 160 hours, so the mean \( \mu = 160 \) hours. Given that \( \lambda = \frac{1}{\mu} \), therefore \( \lambda = \frac{1}{160} \).
2. **Probability Calculation**: The probability that the generator is operational between 60 and 160 hours can be calculated using the cumulative distribution function (CDF) of the exponential distribution:
\[
F(x; \lambda) = 1 - e^{-\lambda x}
\]
Therefore, you find the probability at 160 hours and subtract the probability at 60 hours.
For an educational website, it is important to follow through step-by-step calculations and perhaps include interactive tools or simulations for better understanding.
### Navigation:
- Previous
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