Probability

Question 1

Probability

What is the distribution function of a random variable X (D)

Answer 1

F(x)=P(X<=x)

Question 2

Probability

Define the median of a distribution X (D)

Answer 2

The point m where P(X<=m)>=1/2 and P(X>=m)>=1/2. Or if this occurs over an interval, the midpoint of that interval

Question 3

Probability

What is the definition of convergence in distribution? (D)

Answer 3

For all point x where F is continuous, Fn(x)->F(x)

Question 4

Probability

What is the definition of convergence in probability? (D)

Answer 4

For all ε>0,

P( ¦Xn-X¦ > ε ) -> 0 as n->∞

Question 5

Probability

What is the definition of convergence almost surely? (D)

Answer 5

P ( Xn -> X ) =1

So for all ε>0 there exists an N where ¦Xn-X¦

Question 6

Probability

What is that annoying proof you need to prove the convergence of random variables proofs> (T)

Answer 6

If An is a sequence of increasing events. A1 in A2 in A3 (A1 has the smallest probability) then
lim[n->∞] P (An) = P ( U An )

Question 7

Probability
What is the rough proof of proving that:
For an increasing sequence of events An
lim[n->∞] P (An) = P ( U An ) (T)

Answer 7

Start on RHS. Write as a disjoint set then write as a sum rather than union. Write the sum as a limit. Take the limit outside then put the sum back into a union.

Question 8

Probability

Give the rough proof that convergence in probability => convergence in distribution (T)

Answer 8

Note if Xn<= x then either X<=x+ε or ¦Xn-X ¦>ε but that these probabilities overlap.

So Fn(x)<=P(X<=x+ε or ¦Xn-X¦>ε) then show Fn(x)F(x-ε)-ε so Fn(x)->F(x)

Question 9

Probability

Give the rough proof that convergence almost surely => convergence in probability (T)

Answer 9

Let AN = { ¦Xn-X¦=N }. Almost surely guarantees AN for some N. So P ( U An ) = 1 = lim[n->∞] P(An) by weird lemma. But An => ¦Xn-X¦

Question 10

Probability

Give an example as to why convergence in probability =/=> convergence almost surely (T)

Answer 10

Let Xn be a variable st. P(X=1)=1/n. P(X=0)=(n-1)/n.
Then P ( ¦Xn-0¦= P( Xn = 0 ) -> 1
So Xn->0 in probability. Since Xn discrete {Xn->0}={Xn=0 eventually}={U Bn}

Let BN = { Xn = 0 for all n>=N } . Then for any N,K>0
P(BN)<= P ( Xn = 0 for all n=N,…,N+K )
= N-1/N * N/N+1 * … * N+K-1/N+K = N-1/N+K
Since K arbitrary P(BN)=0

So lim[n->∞] P (Bn) = 0 = P( U Bn ) so Xn-/->0 a.s

Question 11

Probability

Give an example as to why convergence in distribution =/=> convergence in probability (T)

Answer 11

Let Y be st. P(Y=0)=1/2=P(Y=1`) and let Xn have the same distribution as Y but that P(Xn=Y)=0 for all n
Xn clearly -> Y in distribution but P ( ¦Xn-Y¦>1/2 ) = 1 for all n so no convergence in probability

Question 12

Probability

Under what conditions does convergence in distribution => convergence in probability (T)

Answer 12

When Xn converges to a constant c

Question 13

Probability

State Markov’s inequality (T)

Answer 13

For non-negative X and z>0

P(X>=z)<=E[X]/z

Question 14

Probability

Prove Markov’s inequality (T)

Answer 14

Let Xz be st.
Xz=0 when 0<=X=z

So X>=Xz

E[X]>=E[Xz]= 0*P(X=z) then rearrange

Question 15

Probability

State Chebyshev’s inequality (T)

Answer 15

For Y with finite mean and variance and ε>0

P ( ¦Y-E[Y]¦>=ε ) <= Var(Y)/ε^2

Question 16

Probability

Prove Chebyshev’s inequality (T)

Answer 16

P( ¦Y-E[Y]¦>=ε ) = P( (Y-E[Y])^2 >= ε^2 ) <= E[ (Y-E[Y])^2 ]/ε^2 by Markov = Var(Y)/ε^2

Question 17

Probability

State the weak law of large numbers (T)

Answer 17

Let Xi be i.i.d with mean μ and Sn=X1+…+Xn

Then Sn/n -P-> μ

Question 18

Probability

Prove the weak law of large numbers (T)

Answer 18

E[Sn/n]=μ by independence. Var(Sn/n)=Var(Sn)/n^2=( Var(X1)+Var(X2)+…+Var(Xn) ) / n^2 = nσ/n^2=σ/n

By Chebyshev P ( ¦Xn-μ¦>ε ) <= Var(Sn/n)/ε^2 = σ/nε^2 -> 0 as n->∞

Question 19

Probability

State the strong law of large numbers (T)

Answer 19

Let Xi be i.i.d with mean μ and Sn=X1+…+Xn

Then Sn/n -> μ almost surely

Question 20

Probability

State the central limit theorem (T)

Answer 20

Let Xi be i.i.d with mean μ and variance σ. Let Sn=X1+…+Xn

Then Sn-μ/σroot(n) -d-> N(0,1)

Question 21

Probability

What 3 things does the CLT tell us? (Q)

Answer 21

1. That the distribution of Sn concentrates around nμ
2. That the fluctuations of Sn are of order root(n)
3. That the asymptotic distribution of these fluctuations are normal