I am currently trying to solve some exercises in machine-learning/ gaussian distributions.
I have the formula of a multivariate gaussian density function, given mu and covariance matrix. $\mu$ is, in this case, $0$.
$X_1$ and $X_2$ are random variables, the dimension of the density function is $2.$
$$x=(X_1, X_2)$$
$$\Sigma = \begin{bmatrix}\sigma^2 & \alpha\sigma^2 \\ \alpha\sigma^2 & \sigma^2 \end{bmatrix}$$
$$ f(x\mid\mu,\sigma) =\frac{1}{\sqrt{(2\pi)^{2}|\Sigma|}} \exp(-\frac{1}{2}(x)^t\Sigma^{-1}(x))$$
The first question is to insert the given parameters and calculate joint probability $p(X_1, X_2)$.
Here is my question: Is the formula with mü and cov inserted already $p(X_1, X_2)$, or do I need to double integrate over $X_1, X_2$ to calculate the CDF.
The next question would be to calculate $p(X_1)$ ... do I then need to integrate again over $X_2$?
I tried it with online integration, and the results are really, really big.
I hope someone can help me :)
You need to integrate $f$ over $(X_{1},X_{2})$ so you basically have to solve for the double integral.
You need to integrate $f$ for $X_{2}\in\mathbb{R}$ and you will obtain the marginal distribution for $X_{1}$, which will also be gaussian.