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Questions and Answers
What are the parameters of the gamma variable that models the claim amount X?
What are the parameters of the gamma variable that models the claim amount X?
Which moment generating function corresponds to a gamma variable with parameters α=3 and λ=1/4?
Which moment generating function corresponds to a gamma variable with parameters α=3 and λ=1/4?
What is the expression derived from using moment generating functions in the problem?
What is the expression derived from using moment generating functions in the problem?
To find the probability that a claim amount exceeds £20,000, what must you determine about the gamma distribution?
To find the probability that a claim amount exceeds £20,000, what must you determine about the gamma distribution?
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If £20,000 corresponds to a certain z-score, what is the z-score formula for gamma distributed variables?
If £20,000 corresponds to a certain z-score, what is the z-score formula for gamma distributed variables?
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Study Notes
Gamma Distribution Parameters
- Claim amount X is modeled using a gamma distribution with shape parameter α=3 and rate parameter λ=1/4.
- The probability density function for a gamma variable is defined, which involves the parameters α and λ.
Moment Generating Functions
- Moment generating functions (MGFs) are used to calculate moments of the distribution.
- For a gamma distribution, the MGF is given by the formula: M(t) = (1 - λt)^{-α}, valid for t < 1/λ.
- The second moment (E[X^2]) can be calculated using the second derivative of the MGF evaluated at t=0.
Claim Amount Calculation
- To show that ( \frac{1}{4}X^2 - X^2 ), differentiate the MGF and calculate expected values accordingly.
- Utilize properties of MGFs to derive relationships between moments.
Probability Calculation
- To find the probability that a claim amount exceeds £20,000, convert this value to units of the gamma variable (in thousands of pounds).
- Calculate the threshold: for X exceeding £20, use the relationship (X > 20) in terms of the gamma distribution.
- Access and utilize gamma distribution tables to find cumulative probabilities for the calculated values.
Conclusion
- The use of the gamma distribution is essential in modeling claim amounts for industrial policies.
- Applying moment generating functions provides insight into the variability and expected outcomes of claims.
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Description
Quiz on gamma distribution parameters and moment generating functions. Learn about the probability density function, calculating moments, and formulae for MGF.
