Below is a problem which I did and I got the answer in the book but it seemed too easy. Is there something I am missing?
Bob
Problem:
Let $(X,Y)$ be a bivariate normal r.v. where $X$ and $Y$ each have zero mean
and variance $\sigma^2$, and the correlation coefficient of $X$ and $Y$ is
$\rho$. Find the joint pdf of $(X,Y)$.
Answer:
The general form of a bivariate normal distribution of two variable is:
\begin{eqnarray*}
f_{XY}(x,y) &=&
\frac{1}{2\pi \sigma_x \sigma_y(1-\rho^2)^{\frac{1}{2} } }e^{-\frac{1}{2}q(x,y) } \\
\text{ where } \, \\
q(x,y) &=&
\frac{1}{1-\rho^2} \Big(
(\frac{x - u_x}{\sigma_x})^2 - 2\rho( \frac{x - u_x}{\sigma_x} )
( \frac{y - u_y}{\sigma_y} ) + ( \frac{y - u_y}{\sigma_y} )^2
\Big) \\
\end{eqnarray*}
In this case, we have $\sigma_x = \sigma_y = 1$. We also have $u_x = u_y = 0$.
\begin{eqnarray*}
f_{XY}(x,y) &=&
\frac{1}{2\pi (1-\rho^2)^{\frac{1}{2} } }e^{-\frac{1}{2}q(x,y) } \\
\text{ where } \, \\
q(x,y) &=& \frac{1}{1-\rho^2}(x^2 - 2 \rho x y + y^2) \\
\end{eqnarray*}