Let X 1 , X 2 X_{1}, X_{2} X 1 , X 2
f X 1 , X 2 ( x 1 , x 2 ) = 1 2 π σ 1 σ 2 1 − ρ 2 exp [ − φ ( x 1 , x 2 ) 2 ( 1 − ρ 2 ) ] f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right)=\frac{1}{2 \pi \sigma_{1} \sigma_{2} \sqrt{1-\rho^{2}}} \exp \left[-\frac{\varphi\left(x_{1}, x_{2}\right)}{2\left(1-\rho^{2}\right)}\right] f X 1 , X 2 ( x 1 , x 2 ) = 2 π σ 1 σ 2 1 − ρ 2 1 exp [ − 2 ( 1 − ρ 2 ) φ ( x 1 , x 2 ) ]
where
φ ( x 1 , x 2 ) = ( x 1 − μ 1 σ 1 ) 2 − 2 ρ ( x 1 − μ 1 σ 1 ) ( x 2 − μ 2 σ 2 ) + ( x 2 − μ 2 σ 2 ) 2 \varphi\left(x_{1}, x_{2}\right)=\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}-2 \rho\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)+\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)^{2} φ ( x 1 , x 2 ) = ( σ 1 x 1 − μ 1 ) 2 − 2 ρ ( σ 1 x 1 − μ 1 ) ( σ 2 x 2 − μ 2 ) + ( σ 2 x 2 − μ 2 ) 2
and x 1 , x 2 ∈ R x_{1}, x_{2} \in \mathbb{R} x 1 , x 2 ∈ R
(a) Deduce that the marginal probability density function
f X 1 ( x 1 ) = 1 2 π σ 1 exp [ − ( x 1 − μ 1 ) 2 2 σ 1 2 ] f_{X_{1}}\left(x_{1}\right)=\frac{1}{\sqrt{2 \pi} \sigma_{1}} \exp \left[-\frac{\left(x_{1}-\mu_{1}\right)^{2}}{2 \sigma_{1}^{2}}\right] f X 1 ( x 1 ) = 2 π σ 1 1 exp [ − 2 σ 1 2 ( x 1 − μ 1 ) 2 ]
(b) Write down the moment-generating function of X 2 X_{2} X 2 μ 2 \mu_{2} μ 2 σ 2 ⋅ [ N o \sigma_{2} \cdot[N o σ 2 ⋅ [ N o
(c) By considering the ratio f X 1 , X 2 ( x 1 , x 2 ) / f X 2 ( x 2 ) f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right) / f_{X_{2}}\left(x_{2}\right) f X 1 , X 2 ( x 1 , x 2 ) / f X 2 ( x 2 ) X 2 = x 2 X_{2}=x_{2} X 2 = x 2 X 1 X_{1} X 1 μ 1 + ρ σ 1 ( x 2 − μ 2 ) / σ 2 \mu_{1}+\rho \sigma_{1}\left(x_{2}-\mu_{2}\right) / \sigma_{2} μ 1 + ρ σ 1 ( x 2 − μ 2 ) / σ 2 σ 1 2 ( 1 − ρ 2 ) \sigma_{1}^{2}\left(1-\rho^{2}\right) σ 1 2 ( 1 − ρ 2 )