Solutions to the Problems from 3.5-3.7

3.56

Facts You Need to Use:

  1. P(X=1, Y=2)=f(1,2)=1/20
  2. P(X=0,1_Y<3)=f(0,1)+f(0,2)=1/4+1/8=3/8
  3. P(X+Y_1)=f(0,0)+f(0,1)+f(1,0)=1/6+1/4+1/12=1/2
  4. P(X>Y)=f(1,0)+f(2,0)+f(2,1)=1/6+1/24+1/40=28/120=7/30

3.57

Facts You Need to Use:

  1. F(1.2,0.9)=P(X_1.2,Y_0.9)=f(0,0)+f(1,0)=1/6+1/12=1/4
  2. 0
  3. 1/12+1/6+1/24=7/24
  4. F(4,2.7)=1-f(0,3)=1-1/120=119/120

3.63

Facts You Need to Use:

3.76

Facts You Need to Use:

 

There are 2x3x2=12 combinations of x,y,z

 

3.77

  1. P(X=1,Y_2,Z=1)=f(1,1,1)+f(1,2,1)=1/18
  2. P(X=2,Y+Z=4)=14/54=7/27

3.87

Facts You Need to Use:

3.89

  1. g(-1)=1/4, g(1)=3/4, 0 otherwise
  2. h(-1)=5/8, h(0)=1/4, h(1)=1/8, 0 otherwise

 

3.90

3.105

a. Approach 1: Get the marginal distribution of X first then integrate over the appropriate region. Approach 2:

3.108

Facts You Need to Use:

If X and Y are independent iff f(x,y)=g(x)h(y) for all x and y.