Question

Create a pair of random variables, where the correlation between X_1and X_2 is small. We can do this by letting X_2=a*X_1+b*Z, (X_1 and Z be independent random variables) and changing the constant values of a and b.

Answer 1

Let ~ N(0,1) and Z ~ N(0,1) and and Z are independent.

Let = 0.1 + 0.9Z

Thus, E() = 0.1 * E() + 0.9 E(Z) = (0.1 * 0) + (0.9 * 0) = 0.

Var() = [ Cov(X2, Z) = 0 as X2 and Z are independent]

i.e. Var() = 0.01 + 0.81 = 0.82.

Hence, ~ N(0, 0.82).

Now, Cov(​​, ) = Cov(0.1 + 0.9Z, ) = 0.1Cov(, ) + 0.9 Cov(Z, ) = 0.1 + 0 = 0.1.

Hence, correlation between and is small.