(a) Find Cov(4 + 2X , 3 − 2Y ). (b) Let Z = 3X − 2Y + 2. Find E[Z] and σ2z (c) Calculate the correlation coefficient between X and Y. What does this suggest about the relationship between X and Y?
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Let X and Y have the following joint distribution:
X \Y 0 1
0 0.40 0.10
1 0.10 0.10
2 0.10 0.20
(a) Find Cov(4 + 2X , 3 − 2Y ).
(b) Let Z = 3X − 2Y + 2. Find E[Z] and σ2z
(c) Calculate the
(d) Show that for two nonzero constants a and b, Cov(X + a, Y + b) = Cov(X , Y ).
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- The joint probability mass function p(x,y) of X and Y is defined as follows: p(0,0)=0.5 p(1,0)=0.2 p(0,1)=0.2 p(1,1)=0.1 a) What is the covariance between X and Y? b) What is the correlation coefficient of X and Y? c) Are X and Y are dependent? What about are X and Y are uncorrelated?Let X and Y be random variables, and a and b be constants. a) Prove that Cov(aX, bY) = ab Cov(X,Y). b) Prove that if a > 0 and b > 0, then ρaX,bY = ρX,Y. Conclude that the correlation coefficient is unaffected by changes in units.Let X and Y be random variables, and a and b be constants. ???? a) Show that Cov [aX,bY] = abCov [X,Y] . b) Show that if a > 0 and b > 0, then the correlation coefficient between aX and bY is the same as the correlation coefficient between X and Y . c) Is the correlation coefficient between X and Y unaffected by changes in the units of X and Y ?
- Let Z~Normal(0,1). Define X1 = Z and X2 = Z2. Calculate the Spearman correlation coefficient for the pair (X1, X2).1.Calculate the Pearson correlation coefficient. (you only need to show the values of SSxy, SSx and SSy in the formula, and the value of r. ) 2.Is this Pearson’ r significant at level of αtwo-tail = 0.05? (you only need to show the critical value of r, and the comparison between robt and rcrit.) ID X Y S01 20 190 S02 25 177 S03 30 183 S04 35 153 S05 40 159 S06 45 155 S07 50 130 S08 55 122 S09 60 126 S10 60 105Let X be a random variable and Y = aX + b with a ̸= 0. Show that the correlation between X and Y is either +1 or −1. State the conditions which make the correlation equal to +1.
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