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### Introduction to Weibull Probability Distribution

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• 2. Many probability distributions are not a single distribution but are in fact a family of distributions. A family of probability distributions refers to a group of related probability distributions that share certain common properties but differ in specific parameters. These parameters allow for flexibility in shaping the distribution to accurately model different types of data.
• 3. The importance of families of probability distributions: Versatility: They offer a wide range of shapes and properties to fit diverse data sets. Parsimony: They avoid the need to develop a new distribution for every unique data set. Interpretation: Different family members often share interpretations for their parameters, facilitating understanding. Statistical inference: Well-established statistical methods exist for estimating parameters and testing hypotheses within a family.
• 4. Selected Family Probability distribution : Weibull Distribution – 2 Parameter X – Time to failure data X ∼ Weibull(α, β)
• 5. Weibull Distribution: Definition and Characteristics  The Weibull distribution is a two-parameter continuous probability distribution.  It is characterized by two parameters: Shape parameter (α): Determines the skewness and tail behavior of the distribution. Scale parameter (β): Determines the location and spread of the distribution.
• 6. Flexibility: Can model both monotonic increasing and decreasing hazard rates (failure rate over time). Asymptotic behavior: Converges to an exponential distribution for small α and a Rayleigh distribution for large α. Different combinations of α and β result in diverse shapes:  α < 1: Decreasing hazard function, "early failures" more likely.  α = 1: Exponential distribution, constant hazard function.  α > 1: Increasing hazard function, "late failures" more likely.
• 7. Specific Distributions under the Weibull Family  Exponential distribution (α = 1): A special case with constant hazard rate.  Gumbel distribution (α = 1, β = 1): Models extreme values, such as maximum daily rainfall.  Rayleigh distribution (α = 2): Models circular data, such as wind directions.  Weibull-extreme value distribution: Combines Weibull and Gumbel characteristics for extreme value modeling.  Weibull-Fréchet distribution: A generalized form including a location parameter, useful for shifted life-time data.
• 8. Practical Applications of the Weibull Distribution  Reliability engineering: Estimating component lifetimes, predicting failure times, setting maintenance schedules.  Meteorology: Modeling wind speeds, rainfall patterns, forecasting extreme weather events.  Biology: Studying survival times of organisms, analyzing growth rates.  Survival analysis: Modeling time to death or event occurrence in biological or medical studies.  Economics and finance: Analyzing income distribution, modeling financial market returns.  Engineering and science: Analyzing material strengths, fracture times, fatigue crack growth.
• 9. Get Weibull probability distribution under different parameters  Simulates 1000 random draws from a Weibull distribution with a shape parameter (α) of 1 and a scale parameter (β) of 3.
• 10. Weibull(α=1, β=3) • The histogram provides a visual representation of the distribution of the simulated data. Since the shape parameter is 1, the distribution is exponential-like, and you might observe a skewed, right-tailed distribution. • The red curve overlaid on the histogram represents the theoretical probability density function (PDF) of the Weibull distribution with the specified shape and scale parameters. In this case, with a shape of 1, it resembles an exponential distribution.
• 11. Weibull(α=4, β=3) • The histogram provides a visual representation of the distribution of the simulated data. With a shape parameter of 4, the distribution is more peaked and may have a heavier tail compared to distributions with lower shape parameters. And also, it is seemed to like symmetric. • The red curve overlaid on the histogram represents the theoretical probability density function (PDF) of the Weibull distribution with the specified shape and scale parameters. In this case, with a shape of 4, the distribution may exhibit more pronounced skewness and a higher probability of observing larger values.
• 12. Weibull(α=8, β=3) • With a shape parameter of 8, the distribution is likely to be more peaked and have a higher probability of larger values compared to distributions with lower shape parameters. And it also seemed to be left skewed. • The red curve overlaid on the histogram represents the theoretical probability density function (PDF) of the Weibull distribution with the specified shape and scale parameters. In this case, with a shape of 8, the distribution may exhibit a more pronounced peak and a higher probability of extreme values.
• 13. Weibull(α=1, β=100) • With a shape parameter of 1, the distribution is exponential-like, and the scale parameter of 100 indicates that the values are spread out over a larger range. And also, it is right skewed. • The red curve overlaid on the histogram represents the theoretical probability density function (PDF) of the Weibull distribution with the specified shape and scale parameters. In this case, with a shape of 1 and a scale of 100, the distribution may resemble an exponential distribution with a longer tail.
• 14. Conclusion In summary, we can interpret the results by understanding the characteristics of the simulated data, the shape of the histogram, and the overlaying PDF curve, considering the impact of a larger scale parameter on the spread of values in the distribution. And also, we can get different values to the α,β to get various distributions in Weibull family.
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