Gamma Distribution and Moment Generating Functions

Questions and Answers

α=3, λ=1/4 (correct)

α=2, λ=1/2

α=1, λ=1/5

α=4, λ=1/3

M(t) = (1 - 4t)^{-2}

M(t) = (1 - rac{4t}{3})^{-3} (correct)

M(t) = (1 - t)^{-3/4}

M(t) = (1 - 4t)^{-3}

$rac{1}{2}X + X^2$

$rac{1}{4}X^2 - X^2$ (correct)

$rac{1}{2}X^2 + X^2$

$rac{1}{4}X^2 + X^2$

<p>The cumulative distribution function</p> Signup and view all the answers

<p>$z = rac{X - ext{mean}}{ ext{std dev}}$</p> Signup and view all the answers

Study Notes

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 (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.

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.

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.

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.