Question

Consider the following joint PMF of the random variables X and Y:

pX,Y(x,y)={1/72⋅(x2+y2),if x∈{1,2,4} and y∈{1,3}, 0, otherwise}.

1. P(Y<X)=

2. P(Y=X)=

3. Find the marginal PMF pX(x).

pX(1)=
pX(2)=
pX(3)=
pX(4)=

4. Find E[X] and E[XY].

E[X]=
E[XY]=

5. var(X)=

6. Let A denote the event X≥Y. Find E[X∣A].

E[X∣A]=

Answer 1

1.47/22
2.1/36
3a.12/72
3b.1/4
3c.0
3d.42/72
4.3
5.???
6.1.5
7.??

Answer 2

1???
2 2/72
3a 12/72
3b 1/4
3c 0
3d 42/72
4a 3
4b 61/9
5 3/2
6 173/47

Answer 3

1. 47/72

Answer 4

c= 5/64

P(Y<X)= 83/128

P(Y=X)= 1/32

P(X=1)= 10/64

P(X=2)= 17/64

P(X=3)= 0

P(X=4)= 37/64

E[X]= 3

E[XY]= 227/32

var(X)= 3/2

Answer 5

c= 1/128

P(Y<X)= 83/128
P(Y=X)= 1/32
P(X=1)= 10/64
P(X=2)= 17/64
P(X=3)= 0
P(X=4)= 37/64
E[X]= 3
E[XY]= 227/32
VAR(X)= 3/2