Paper 2, Section I, F

Probability | Part IA, 2009

Consider a pair of jointly normal random variables X1,X2X_{1}, X_{2}, with mean values μ1\mu_{1}, μ2\mu_{2}, variances σ12,σ22\sigma_{1}^{2}, \sigma_{2}^{2} and correlation coefficient ρ\rho with ρ<1|\rho|<1.

(a) Write down the joint probability density function for (X1,X2)\left(X_{1}, X_{2}\right).

(b) Prove that X1,X2X_{1}, X_{2} are independent if and only if ρ=0\rho=0.

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