Suppose that Rich is the third in line. Given that Rich has been waiting t minutes, find the distribution of the waiting time for being the first in line. Find the conditional mean. (Advice: The reader who has solved Exercise 51 may just apply the result.
The reader who skipped Exercise 51, may consider this particular case proceeding from the formula (5.1.5) where the role of X₁ should be played by S₂, the role of X₂ by X₃, and the total S = S₃. For the densities of S₂ and S₃, we may use Proposition 6.)

Exercise 51

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].