Consider the exponential family of distribution in which the probability density/mass function takes the form f(y;0,0)=...

 

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Consider the exponential family of distribution in which the probability density/mass function takes the form f(y;0,0)= exp (30-b(0)]/a(o) + c(y, o)}. The probability density function of the gamma distribution is 178 8(1:14, 6) = T(1/8) () exp(-34). (a) Show that the gamma distribution belongs to exponential family, and also identify o, 0, b(0), a(o), and c(y, o). (b) Find the mean and variance of a gamma random variable in terms of u and 3. (8 marks) (4 marks) (2 marks) (c) Find the canonical link function. (d) Determine whether the following functions are suitable to be used as the link functions in gamma regression, and motivate your answer. (1) g() = -2 (ii) g() = 1/ (iii) g(u) = log(u) V> 0. (3 marks) (3 marks) (3 marks) Consider the exponential family of distribution in which the probability density/mass function takes the form f(y;0,0)= exp (30-b(0)]/a(o) + c(y, o)}. The probability density function of the gamma distribution is 178 8(1:14, 6) = T(1/8) () exp(-34). (a) Show that the gamma distribution belongs to exponential family, and also identify o, 0, b(0), a(o), and c(y, o). (b) Find the mean and variance of a gamma random variable in terms of u and 3. (8 marks) (4 marks) (2 marks) (c) Find the canonical link function. (d) Determine whether the following functions are suitable to be used as the link functions in gamma regression, and motivate your answer. (1) g() = -2 (ii) g() = 1/ (iii) g(u) = log(u) V> 0. (3 marks) (3 marks) (3 marks)