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Covariance & Correlation

  • Covariance describes the co-movement between 2 random numbers, given as:
    • Cov(X1, X2) = σ12

  • Correlation coefficient is a unit-less number, which gives a measure of linear dependence between two random variables.
    • ρ(X1, X2) = Cov(X1, X2)/ σ1σ2
  • Correlation coefficient always lies in the range of +1 to -1
  • A correlation of 1 means that the two variables always move in the same direction
  • A correlation of -1 means that the two variables always move in opposite direction
  • If the variables are independent, covariance and correlation are zero, but vice versa is not true

Given two random variables X and Y, what is the Variance of X given Variance[Y] = 100, Variance [4X - 3Y] = 2,700 and the correlation between X and Y is 0.5?

  1. 56.3
  2. 113.3
  3. 159.9
  4. 225.0

Using the theorems on variance and covariance
Variance [4X-3Y] = 16*Var[X] + 9*Var[Y] + 2*4*(-3)*Var[X]^(1/2)* Var[Y]^(1/2)*correlation[X,Y]
Solve for Var[X] = 225.0


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