Let X and Y have the following joint distribution a Find the

Let X and Y have the following joint distribution:

(a) Find the probability distributions for X and Y .

(b) Find E[X] and E[Y].

(c) Find the probability that X is larger than 1.

(d) Find E[XY]

(e) Find the covariance between X and Y

(f) Using your answer from part (e), are X and Y independent? Explain

Solution

(a) Find the probability distributions for X and Y .

P(X=0)=0.20+0.15=0.35

P(X=2)=0.10+0.2=0.3

P(X=4)=0.25+0.1=0.35

P(Y=-1)=0.20+0.1+0.25 =0.55

P(Y=1)=0.15+0.2+0.1=0.45

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(b) Find E[X] and E[Y].

E(X)=sum of x*f(x)

=0*0.35+2*0.3+4*0.35 =2

E(Y)=sum of y*f(y)

=-1*0.55+1*0.45

=-0.1

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(c) Find the probability that X is larger than 1.

P(X>1)=P(X=2)+P(X=4)=0.3+0.35 =0.65

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(d) Find E[XY]

=sum of xy*f(xy)

=0*(-1)*0.2+0*1*0.15+2*(-1)*0.1+2*1+0.2+4*(-1)*0.25+4*1*0.1

=1.4

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(e) Find the covariance between X and Y

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

=1.4-2*(-0.1)

=1.6

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(f) Using your answer from part (e), are X and Y independent? Explain

X and Y are not independent because P(X=0)*P(Y=-1) is not equal to P(X=0 and Y=-1)