Answer :
Final answer:
The expected number of years before a solar panel fails is 10 years. The probability that none of the 8 solar panels will fail in the first 2 years is approximately 10.07%. If 10,000 solar panels are sold, the expected amount of money paid in warranty claims would be $2,000,000.
Explanation:
The exponential distribution is commonly used to model the longevity of devices, such as solar panels. In this case, the exponential distribution has a probability density function with λ = 0.1 per year. The mean or average time before a solar panel fails can be calculated as 1/λ. Therefore, the expected number of years before a solar panel fails is 1/0.1 = 10 years.
To find the probability that none of the 8 solar panels will fail in the first 2 years, we can use the exponential distribution. The probability that a single solar panel will not fail in the first 2 years is given by P(X > 2) = e^(-λt) = e^(-0.1*2) = e^(-0.2) ≈ 0.8187. Since each solar panel is independent, the probability that none of the 8 panels will fail in the first 2 years is (0.8187)^8 ≈ 0.1007, or approximately 10.07%.
If each warranty claim costs $2000 and you sold 10,000 solar panels, you can expect to pay in warranty claims a total amount of 10,000 * 0.1 * 2000 = $2,000,000. The expected amount of money you would pay in warranty claims can be calculated by multiplying the number of solar panels sold, the failure rate (λ), and the warranty claim cost.
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