Q 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter a and rate parameter 3. Given Y = y, the number of accidents that the individual suffers in years 1,2,..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X1, Xn can be described via P(a ≤ Y ≤ b, X₁ = k1, X2 = k2 . . . Xn = kn) = f* f (y, k₁, k2, ... kn) dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X₁, X₂,... Xn are independent, each having the Poisson distribution with parameter y.
Q 4.4. An individual picked at random from a population has a propensity to have accidents that is modelled by a random variable Y having the gamma distribution with shape parameter a and rate parameter 3. Given Y = y, the number of accidents that the individual suffers in years 1,2,..., n are independent random variables X₁, X2,... Xn each having the Poisson distribution with parameter y. (a) Write down a function f so that the joint distribution of Y, X1, Xn can be described via P(a ≤ Y ≤ b, X₁ = k1, X2 = k2 . . . Xn = kn) = f* f (y, k₁, k2, ... kn) dy and derive from this expression that, for your choice of f, Y has the Gamma distribution, and that conditionally on Y = y, X₁, X₂,... Xn are independent, each having the Poisson distribution with parameter y.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
Related questions
Question
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 5 steps with 47 images
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,