Let X, X2,.. X be i.i.d. continuous random variables with E(X) = 0, Var(X) = 1....

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Let X₁, X2,.. X be i.i.d. continuous random variables with E(X₁) = 0, Var(X₁) = 1. Let Z be a discrete random variable independent of the X,'s with the following pmf: if z = -1 Pz(z)= if z=0 if z = 1 For each i=1,2,..., n let Y, X, Z. a. Show that the Y's are not independent, but are uncorrelated; recall that for random variables X and Y, Correlation (X,Y)= Covariance (X,Y), Var(X)Var(Y) correlation indicates the strength of the linear relationship between the two variables. Uncorrelated random variables have correlation equal to 0. (Hint: For independence, compare P(Y, = 0, Y = 0) to P(Y, = 0)P(Y, =0)) Let X₁, X2,.. X be i.i.d. continuous random variables with E(X₁) = 0, Var(X₁) = 1. Let Z be a discrete random variable independent of the X,'s with the following pmf: if z = -1 Pz(z)= if z=0 if z = 1 For each i=1,2,..., n let Y, X, Z. a. Show that the Y's are not independent, but are uncorrelated; recall that for random variables X and Y, Correlation (X,Y)= Covariance (X,Y), Var(X)Var(Y) correlation indicates the strength of the linear relationship between the two variables. Uncorrelated random variables have correlation equal to 0. (Hint: For independence, compare P(Y, = 0, Y = 0) to P(Y, = 0)P(Y, =0))

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