Review

Question 1

Define E[X] and E[Z|Y] through the tower rule

Answer 1

E[X] = E[E[X|Y ]]

E[Z|Y ] = E[E[Z|X]|Y]

Question 2

State Markov’s inequality

Answer 2

P( |X| >= t) <= E[ |X| ] / t

Question 3

State Slutsky’s theorem

Answer 3

If Xn –d–> X and Yn –p–> c then
Xn + Yn –d–> X + c
XnYn –d–> cX
Xn/Yn –d–> Xn/c

Question 4

What is the continuous mapping theorem?

Answer 4

Xn –d–> X implies g(Xn) –d–> g(X)

Xn –P–> X implies g(Xn) –P–> g(X)

Question 5

State the weak law of large numebrs

Answer 5

For Xi iid with E|Xi| < inf we have that as n -> inf

mean(Xn) –P–> E[X1]

Question 6

State the central limit theorem

Answer 6

as n -> inf

sqrt(n) ( mean(X) - E[X1]) –d–> N(0, var(X))

Question 7

What does it mean that Xn converges to X in distribution? And probability?

Answer 7

Distribution: F_{Xn}(x) → F_{X}(x)
Probability:for any E>0, P(|Xn − X| ≥ E) →0 as n → ∞