Verify the entries in Table 13.8 for the gamma distribution. Specifically:
a. Show that the gamma is a member of the linear exponential family of distributions.
b. Describe the components of the linear exponential family \((\theta, \phi, b(\theta), S(y, \phi))\) in terms of the parameters of the gamma distribution.
c. Show that the mean and variance of the gamma and linear exponential family distributions agree.

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Table 13.8 Selected Distributions of the One-Parameter Exponential Family Density or Mass Function Distribution Parameters General , y0-b(0) exp Normal , Binomal +S(y,)) exp(-) Components 8, o, b(0), S(y,) 92 Ey b'(0) +I(2)) 0 = fl ()'(1-7)- Poisson exp(-) Negative Binomial" (y+r) r. p P'(1-p) (y+r) y!F(r) Gamma , y-exp(-yy) In(), 1, n ln(1+e"), In () In, 1, e, In(y!) In(1 - p), 1,-r In(1-e), In --In(-), In = N e" = r(1-p) P -1=% -In (F())+(-1) In y Inverse Gaussian . exp(-) -1/(22), 1/2,-20, (-20)-1/2 0/(y) -0.5 In(2y) Tweedie See Section 13.6 "This assumes that the parameter r is fixed but need not be an integer.