Let X and Y be rv s in some sample space n and let Z = X + Y, i.e., for each ω ∈ n, Z(ω) = X(ω) + Y(ω).
(a) Show that the set of ω for which Z(ω) = ±∞ has probability 0.
(b) To show that Z = X + Y is a rv, we must show that for each real number α, the set
{ω ∈ n : X(ω) + Y(ω) ≤ α} is an event. We proceed indirectly. For an arbitrary positive integer n and an arbitrary integerk > 0, let B(n, k) = {ω : X(ω) ≤ k/n} n{Y(ω) ≤ α + (1 - k)/n}. Let D(n) = l
B(n, k) and show that D(n) is an event.
(c) On a two-dimensional sketch for a given α, show the values of X(ω) and Y(ω) for which ω ∈ D(n). Hint: This set of values should be bounded by a staircase function.
(d) Show that {ω : X(ω) + Y(ω) ≤ α} = n D(n). (1.100)
Explain why this shows that Z = X + Y is a rv.
(e) Explain why (d) implies that if Y = X1 + X2 + ··· + Xn and if X1, X2, ... , Xn are rv s, then Y is a rv. Hint: Only one or two lines of explanation are needed.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.