Endterm

Question 1

What can you say about the density frunction of Y=g(x) if g is differentiable and one-to-one?

Answer 1

fy(y) = fx(g-1(y)) * 1/ |g’(g-1(y))|

Question 2

What can you say if g is differenciable and strictly increasing? strictly decreasing?

Answer 2

Fy(y) = Fx(g-1(y))
Fy(y) = 1 - Fx(g-1(y))

Question 3

What is a joint probability mass function (discrete random variable)?

Answer 3

P(k1, k2, … , kn) = P(X=k1, X=k2, … , Xn=kn)

Question 4

What is the marginal probability mass function if Xj?

Answer 4

PXj(k) = Σ (l1, … , lj-1, lj+1, …, ln) p(l1, … , lj-1, k, lj+1, …, ln)

Question 5

How can you find E[g(X1,…,Xn)]?

Answer 5

E[g(X1,…,Xn)]=Σ(k1,…,kn) g(k1,…,kn)*p(k1,…,kn)

Question 6

When are two continuous random variables X and Y jointly continuous?

Answer 6

If joint density function fX,Y st

P((X,Y)∈B) = ∫∫(B) fX,Y(x,y) dx dy

Question 7

What properties does the density function f(x,y) has?

Answer 7

f(x,y)>=0

∫∫(both -oo to +oo) f(x,y) dx dy = 1

Question 8

What is the marginal density function of X?

Answer 8

fX(x) = ∫(-oo to +oo) fX,Y(x,y) dy

Question 9

What is E[g(X,Y)] for continuous random variables?

Answer 9

E[g(X,Y)] = ∫∫(-oo to +oo) g(x,y) * fX,Y(x,y) dx dy

Question 10

What can you say about the joint distribution functions (for discrete and continuous) if X and Y are independent?

Answer 10

if X and Y independent then
P(X∈A, Y∈B) = P(X∈A)*P(Y∈B)
so
discrete : PX,Y(x,y) = PX(x) * PY(y)
continuous : fX,Y(x,y) = fX(x) * fY(y)

Question 11

What is a joint cumulative distribution function?

Answer 11

F(S1,…,Sn) = P(X1<=S1,…,Xn<=Sn)

Question 12

What is the probability mass function of X+Y?

Answer 12

PX+Y(n) = PX * PY(n) = Σ(k) PX(k)PY(n-k) = Σ(m) PX(n-m)PY(m)

Question 13

What is the density function of X+Y?

Answer 13

fX+Y(z) = fXfY(z) =∫(-oo to oo)fX(x)fY(z-x)dx =∫(-oo to oo)fX(z-x)*fY(x)dx

Question 14

What does the linearity of expectation say?

Answer 14

E[g1(x1) + g2(x2) + … + gn(xn)] = E[g1(x1)] + E[g2(x2)] + … + E[gn(xn)]
E(X1+…+Xn) = E(X1) + E(X2) + … + E(Xn)

Question 15

What can you say about expectation if Xn are independent rv

Answer 15

the expectation of the product of functions of independent rv is the product of the expectations of the functions of independent rv

Question 16

What is the covariance?

Answer 16

Cov(X,Y) = E(XY) - E(X)*E(Y)

Question 17

What can you say if Cov(X,Y)>0? <0? =0?

Answer 17

> 0 : X and Y deviate together below or above their means
<0 : X and Y deviate in opposite directions
=0 : X and Y are independent

Question 18

What are the properties of the covariance?

Answer 18

Cov(X,Y)=Cov(Y,X)
Cov(X,X) = Var(X)
Cov(aX+b,Y)=aCov(X,Y)

Question 19

What is the correlation of X and Y?

Answer 19

Corr(X,Y) = Cov(X,Y) / sqrt(Var(X))*sqrt(Var(Y))

-1<=Corr(X,Y)<=1

Question 20

What does the central limit theorem say?

Answer 20

X1, X2,… independent identically distributed and E(Xi)=u, Var(Xi)=s^2 and a<=b
lim(n–>oo) p(a<=X1+…+Xn-nu/s*sqrt(n) <=b)
= ∫(a,b) 1/sqrt(2pi) * e(-1/2 x^2) dx
= o/ (b) - o/(a)

Question 21

How can you find E(X) from the conditional expectation?

Answer 21

E(X) = Σ(i=1 to n) E(X/Bi) * P(Bi)

Question 22

What is the conditional probability mass function of X given Y=y? the expectation? the expectation of X? PX(x)? PX,Y(x,y)?

Answer 22

PX/Y(x/y) = P(X=x/Y=y)= P(X=x,Y=y)/P(Y=y) = PX,Y(x,y) / PY(y)

E(X/Y=y) = Σ(x) x*PX/Y(x/y)

E(X)=Σ(y) E(X/Y=y)*PY(y)

PX(x) = Σ(y) PX/Y(x/y)*PY(y)= Σ(y) PX,Y(x,y)

PX,Y(x,y) = PX/Y(x/y)*PY(y)

Question 23

What is the memoriless property?

Answer 23

P(X>t+s / X>t) = P(X>s)