4. (8 marks) Prove that the marginal distributions of a bivariate normal distribution $f\left(x_{1}, x_{2} ight) $ with mean vector $\boldsymbol{\mu)=\left[\mu_{1}, \mu_{2} ight]^{T}$ and variance- covariance matrix $$ \boldsymbol{\Sigma)=\left[\begin{array}{cc} \sigma_{1}^{2} & \sigma_{12} \ \sigma_{12} &\sigma_{2}^{2} \end{array} ight] $$ are univariate normal distributions, that is, $X_{1} \sim N\left(\mu_{1}, \sigma_{1} ight), X_{2} \sim N\left(\mu_{2}, \sigma_{2} ight) $ SP.AS. 1604