# Covariance & Correlation

- Covariance describes the co-movement between 2 random numbers, given as:
- Cov(X
_{1}, X_{2}) = σ_{12 }

- Cov(X

**Correlation coefficient**is a unit-less number, which gives a measure of linear dependence between two random variables.- ρ(X
_{1}, X_{2}) = Cov(X_{1}, X_{2})/ σ_{1}σ_{2}

- ρ(X
- 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

Example

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?

- 56.3
- 113.3
- 159.9
- 225.0

Solution

**D.**

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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