RVs Mean

Question 1

What is expectation of a discrete RV, and when P(X= inf) > 0

Answer 1

Expectation of a discrete RV is:
Sum X in X(omega) x*P(X=x)
If P(X = inf) > 0, E[X] = inf

Question 2

What is expectation of continuous RV

Answer 2

Expectation of continuous RV is:
E[X] = integral inf down to 0 x*fx(x) dx fx is density

Question 3

What is E(LX + UY) for non-negative RVs

Answer 3

E(LX + UY) for non-negative RVs is
LE(X) + UE(Y)

Question 4

When is P(X < inf)

Answer 4

P(X < inf) when
X >= 0 and E(X) < inf

Question 5

When does P(lim k tends to inf Xk = X) = 1

Answer 5

P(lim k tends to inf Xk = X) =1 when (Xn)n is a sequence of non-negative RVs and is monotone increasing (Xn+1 >= Xn)

Question 6

If sum k=1 to inf E[Xk] < inf, what does this imply

Answer 6

If sim k = 1 to inf E(Xk) < inf, this implies E( sum k = 1 to inf Xk ) < inf
P( sim k = 1 to inf Xk < inf) = 1
P( lim k tends to inf Xk = 0) = 1

Question 7

What is the 1st Borel-Cantelli lemma

Answer 7

1st Borel-Canelli lemma is:
If J1, J2,… is a sequence of events and sim n = 1 to inf P(Jn) < inf, then only a finite number of events Jk will occur.
Exists a random K s.t Jk^c occurs for all k >= K

Question 8

What does the monotone convergence theorem states

Answer 8

Monotone convergence theorem states that:
If Xn is a sequence of non-negative RVs s.t Xn tends to X (Xn+1 >= Xn, Xn tends to X) then
P(Xn+1 >= Xn, limXk n tends to inf = X) = 1 then E(Xn) tends to E(X)

Question 9

What does MCT imply

Answer 9

MCT implies continuity of probability measures