Math Problem Statement
Let T ∼ Weibull(0.5, 2.400).
Find P(T > 5). (Round the final answer to four decimal places.)
Solution
The probability density function (PDF) of a Weibull distribution is given by:
where is the shape parameter and is the scale parameter. However, to compute , we use the cumulative distribution function (CDF), which for a Weibull distribution is:
The probability is complementary to the CDF:
Given and , we want to find :
First, calculate :
Now, calculate :
So, the probability is approximately:
Final Answer:
Would you like more details on this calculation or have any questions?
Here are 5 related questions:
- What is the expected value of a Weibull distribution?
- How does the shape parameter affect the distribution's behavior?
- Can you compute for the same distribution?
- What is the variance of a Weibull distribution?
- How do you find the median of a Weibull distribution?
Tip: The Weibull distribution can model a variety of life data behaviors, especially in reliability and survival analysis. Adjusting the shape parameter can mimic different failure rates.
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Math Problem Analysis
Mathematical Concepts
Weibull Distribution
Probability
Exponential Function
Formulas
Weibull CDF: F(t) = 1 - e^{-(t/λ)^k}
Complementary Probability: P(T > t) = e^{-(t/λ)^k}
Theorems
Weibull Distribution Theorem
Suitable Grade Level
Undergraduate Statistics or Advanced High School (AP Statistics)
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