**Conditional Normal Distribution**

Let us consider a two-dimensional normal distribution of some random vector \(\Big(\begin{array}{x}X_{1}\\X_{2}\end{array}\Big)\). The distribution takes the following (general) form:

\(\Big(\begin{array}{x}X_{1}\\X_{2}\end{array}\Big) \sim N_{2}\left(\boldsymbol{\mu} = \Big(\begin{array}{c} \mu_{1} \\ \mu_{2}\end{array}\Big), \Sigma = \left( \begin{array}{cc} \sigma_{1}^{2} & \sigma_{12} \\\sigma_{21} & \sigma_{2}^{2} \end{array} \right) \right)\),
where \(\boldsymbol{\mu} \in \mathbb{R}^2\) is the vector of the expected values and \(\Sigma\) is the variance-covariance matrix, which is a positive definite and symmetric. The correspoding density function (of the two dimensional normal distrubution) is given by the expression

\(\large{f(\boldsymbol{x}) = \frac{1}{2 \pi |\Sigma|^{1/2}} exp\Big\{ -\frac{1}{2} (\boldsymbol{x} - \boldsymbol{\mu})^{\top} \Sigma^{-1} (\boldsymbol{x} - \boldsymbol{\mu}) \Big\},}\)
for \(\boldsymbol{x} = (x_{1}, x_{2})^{\top} \in \mathbb{R}^{2}\).

This density can be used to derive the marginal distrubution of random variables \(X_{1}\) and \(X_{2}\) or the conditional distrubution of \(X_{1}\) given \(X_{2}\) (or \(X_{2}\) given \(X_{1}\) respectively). In the following we will do both.

For the marginal density of \(X_{1}\) we have \(f(x_{1}) = \int_{\mathbb{R}} f(x_{1}, x_{2}) \mbox{d}x_{2}\) and analogously also for the marginal density of \(X_{2}\) (where we need to integrate the join density wrt the first covariate).

For a simple example with a two dimensional normal distribution the conditional distribution distribution of \(X_{2}\) given \(X_{1} = x_{1}\) is normal again and it holds that

\((X_{2} | X_{1 } = x_{1}) \sim N\Big(\mu_{2} + \frac{\sigma_{21}(x_1 - \mu_1)}{\sigma_{1}^2}, \sigma_{2}^2 - \frac{\sigma_{12}\sigma_{21}}{\sigma_{1}^2}\Big).\)
Now we can apply the formulas given above to obtain the marginal and conditional distributions. Firstly, the library `mvtnorm`

needs to be installed on R. The library is loaded in by running the command:

`library("mvtnorm")`

Let us consider a simple example with two dimensional normal distribution with the zero mean vector \(\boldsymbol{\mu} = (0,0)^\top\) and the variance-covariance matrix \(\Sigma = \left( \begin{array}{cc} 1 & 0.8 \\0.8& 1\end{array} \right)\). We would like to calculate the conditional distribution of \(X_{2}\) given \(X_{1} = 0.7\).

```
Sigma <- matrix(c(1,.8,.8,1), nrow=2) ## variance-covariance matrix
x <- seq(-3,3,0.01)
contour(x,x,outer(x,x,function(x,y){dmvnorm(cbind(x,y),sigma=Sigma)}))
abline(v=.7, lwd=2, lty=2, col = "red")
text(0.75, -2, labels=expression(x[1]==0.7), col = "red", pos = 4)
### conditional distribution of X2 | X1 = 0.7
y <- dnorm(x, mean = 0.8 * 0.7, sd = sqrt(1 - 0.8^2))
lines(y-abs(min(x)),x,lty=2,lwd=2, col = "red")
### marginals
m1 <- m2 <- dnorm(x, 0, 1)
lines(x, m1 - abs(min(x)), lty = 1, lwd = 1)
lines(m2 - abs(min(x)), x, lty = 1, lwd = 1)
```

And we can add some other conditional distribution of (e.g. for \(X_{2} | X_{1} = -1\)) by adding additional lines of an R code:

```
contour(x,x,outer(x,x,function(x,y){dmvnorm(cbind(x,y),sigma=Sigma)}))
abline(v=-1, lwd=2, lty=2, col = "blue")
### conditional distribution of X2 | X1 = - 1
y2 <- dnorm(x, mean = 0.8 * (- 1), sd = sqrt(1 - 0.8^2))
lines(-y2 + max(x),x,lty=2,lwd=2, col = "blue")
```