Question2 3+12=15Marks Part (a): Discuss what is meant by the concept of indepen
ID: 2918143 • Letter: Q
Question
Question2 3+12=15Marks
Part (a):
Discuss what is meant by the concept of independence of twodiscrete random variables.
Part (b):
Given is the joint probability distribution of X and Y.
Y
X
0
1
2
3
g(X)
0
0.05
0.05
0.10
0
0.20
1
0.05
0.10
0.25
0.10
0.50
2
0
0.15
0.10
0.05
0.30
h(Y)
0.10
0.30
0.45
0.15
1.00
Determine the followings.
I. Mean of X and Mean of Y
II. Var (X) andVar (Y)
III. Cov (X, Y)
IV. Correlation coefficientr.
Y
X
0
1
2
3
g(X)
0
0.05
0.05
0.10
0
0.20
1
0.05
0.10
0.25
0.10
0.50
2
0
0.15
0.10
0.05
0.30
h(Y)
0.10
0.30
0.45
0.15
1.00
Explanation / Answer
Given is the joint probabilitydistribution of X and Y.
Y
X
0
1
2
3
g(X)
0
0.05
0.05
0.10
0
0.20
1
0.05
0.10
0.25
0.10
0.50
2
0
0.15
0.10
0.05
0.30
h(Y)
0.10
0.30
0.45
0.15
1.00
Determine the followings.
I. Mean of X and Mean of Y
E(X)=0*0.2+1*0.5+2*0.3=1.1
E(Y)=0*0.1+1*0.3+2*0.45+3*0.15=1.65
II. Var (X) and Var (Y)
E(X^2)=0*0.2+(1^2)*0.5+(2^2)*0.3=1.7
Var(X)= E(X^2) -[E(X)]^2 =1.7-1.1^2=0.49
E(Y^2)=0*0.1+1^2*0.3+2^2*0.45+3^2*0.15=3.45
Var(Y) =E(Y^2) - [E(Y)]^2 =3.45-1.65^2=0.7275
III. Cov (X, Y)
E(XY)=1*1*0.1+1*2*0.25+1*3*0.1+2*1*0.15+2*2*0.1+2*3*0.05=1.9
Cov(X,Y)=E(XY)-E(X)E(Y) = 1.9 -1.1*1.65=0.085
IV. Correlation coefficient r.
r = Cov(X,Y)/ [x*y] =0.085/[sqrt(0.49*0.7275)] = 0.1424
Y
X
0
1
2
3
g(X)
0
0.05
0.05
0.10
0
0.20
1
0.05
0.10
0.25
0.10
0.50
2
0
0.15
0.10
0.05
0.30
h(Y)
0.10
0.30
0.45
0.15
1.00
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