The joint probability mass function of random variables X an

The joint probability mass function of random variables X and Y is defined as: Find the value of Cov(X, Y). Find the value of P{ X LE 2Y }. Find the marginal probability mass function of Y. Find the value of P{ Y GE 2 | X = 4}. Find the value of E[2X - 3 | Y = 1 ].

Solution

The marginal distribution of X is

x   p(x)
3   0.3125
4   0.3750
5   0.3125

The marginal distribution of Y is

y   p(y)
1   0.3125
2   0.3750
3   0.3125

a)E(XY)=8

E(X)=4,E(Y)=2

cov(X,Y)=E(XY)-E(X)E(Y)=8-4*2=0

b)P(X<=2Y)=P(X=3,Y=2)+P(X=3,Y=3)+P(X=4,Y=4)+P(X=4,Y=3)+P(X=5,Y=3)=0.1875+0.0625+0+0.1875+0.0625==0.5

c)The marginal pmf of Y is

y   p(y)
1   0.3125
2   0.3750
3   0.3125

d)P(Y>=2,X=4)=0+0.1875=0.1875

P(X=4)=0.3750

P(Y>=2|X=4)=0.5

e)E(2X-3|Y=1)=3*0.625+5*0.1875+7*0.0625=3.25