Let X 1 and X 2 have the -distributions with the common scale parameter a = 1

Let X₁ and X₂ have the Γ-distributions with the common scale parameter a = 1 and essential parameters ν₁ and ν₂, respectively. Let S = X₁+X₂. Given S = s, the r.v. X₁ takes on values from [0, s].

(a) Using formula (5.1.5), prove that the conditional density

where the Beta-function B(ν₁, ν₂) is defined in (4.2.3)-(4.2.4). 

Prove that the corresponding conditional density of the r.v. Y = 1/s X₁ which takes on values from [0,1], is

and does not depend on s. The last distribution is referred to as the Beta-distribution.

(b) Which distribution are we dealing with in (6.4) when ν₁ = ν₂ = 1. Connect this fact.

(c) Proceeding from (6.4), find the conditional distribution of Y and E{Y |S} for ν₁ = ν and ν₂ = 1. Does it grow when ν is increasing? To what does this distribution and E{Y |S}converge as ν→∞. Give a common-sense interpretation.