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IAI Actuarial Core Principles · Actuarial Statistics · Generalised linear models

A random variable Y has density f(y) = exp{ (yθ − b(θ))/φ + c(y,φ) } with b(θ) = θ²/2 and φ = σ². Which distribution does Y follow, and what is its variance function V(μ)?

Y is normal with mean θ. The mean is b'(θ) = θ and b''(θ) = 1, so the variance function is constant at 1 and the variance equals the dispersion parameter σ². Poisson, gamma and the others have different b(θ) functions.

  1. APoisson, with V(μ) = μ
  2. BNormal with mean θ, with V(μ) = 1Correct
  3. CGamma, with V(μ) = μ²
  4. DBinomial, with V(μ) = μ(1 − μ)
  5. Inverse Gaussian, with V(μ) = μ³

Explanation

With b(θ) = θ²/2, the mean is μ = b'(θ) = θ and b''(θ) = 1, so V(μ) = 1. This is the normal distribution with variance φ = σ². Poisson would need b(θ) = e^θ, giving V(μ) = μ.

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