Let X ∼ Exp(λ) with pdf f(x|λ) = λe−λx and λ ∼ Gamma(a, b) with PDF f(λ) = 1 λa−1e−λ/b where a > 0 is the shape parameter and b > 0 is the scale parameter. That Γ(a)ba is to say, the Poisson parameter λ is no longer a constant number; it is a gamma random variable. Find the marginal distribution of X, i. e., write the PDF of X as a function of a and b. (Note that this marginal distribution is called Lomax distribution.)
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Let X ∼ Exp(λ) with pdf f(x|λ) = λe−λx and λ ∼ Gamma(a, b) with PDF f(λ) = 1 λa−1e−λ/b where a >...
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