Let (X, Y ) have bivariate normal density $f(x, y) = \frac{\sqrt{ab−c^2}}{2π}e^{− \frac{1}{2}(ax^2+by^2+2cxy)}$, find marginal distribution of Y.

Question

Let (X, Y ) have bivariate normal density $f(x, y) = \frac{\sqrt{ab−c^2}}{2π}e^{− \frac{1}{2}(ax^2+by^2+2cxy)}$, how can i calculate marginal distribution of Y from the joint pdf. I know we have to integrate joint pdf $wrt$ x, but i am not able to do it.

Answer 1

Hint: \begin{align} &\exp(-ax^2 + by^2+2cyx) = \exp(\frac{c^2y^2}{a})\times \exp(-(\sqrt ax^2 + \frac{2cy\sqrt{a}x}{\sqrt{a}} + \frac{c^2y^2}{a}))\\ &=\exp(\frac{c^2y^2}{a})\times \exp\bigg(-\bigg(\sqrt{a}x+\frac{cy}{\sqrt{a}}\bigg)^2\bigg) \end{align}