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You need to have basic familiarity with univariate normal distribution, and understand the basic property that linear combinations of normals are also normal.
Suppose that \(Z_1,Z_2\) are independent standard normal \(N(0,1)\) and define \(X_1=Z_1+0.1 Z_2\) and \(X_2=Z_1-0.1 Z_2\). What is the joint distribution of \(X_1,X_2\)?
We know from the basic property that \(X_1\) will be univariate normal, and that \(X_2\) will be univariate normal. However, they will not necessarily be independent because \(Z_1\) and \(Z_2\) were used to compute both. Indeed, you can see that \(X_1\) and \(X_2\) might both be expected to be close to \(Z_1\) (because the 0.1 multiplier on \(Z_2\) is “relatively small”). So when \(X_1\) is big we should expect \(X_2\) will likely be big, and when \(X_1\) is small we should expect \(X_2\) will likely small.
The following code illustrates this: the histograms illustrate both \(X_1\) and \(X_2\) are normal, and the scatterplot of \(X_1\) and \(X_2\) shows they are correlated (and the sample correlation is approximately 0.98).
Z1 = rnorm(1000) Z2 = rnorm(1000) X1 = Z1+0.1*Z2 X2 = Z1-0.1*Z2 hist(X1)
plot(X1,X2, main="scatterplot of (X1,X2)", ylim=c(-4,4), asp=1) #asp=1 sets the scales of X1 and X2 the same
In fact the answer to the question “what is the joint distribution of \(X_1,X_2\)” is they have a “bivariate normal distribution”. Thus the scatterplot shown above shows a scatterplot of 1000 samples from a bivariate normal distribution. The prefix “bi” means 2, referring to the fact that here we are looking at 2 variables, \(X_1\) and \(X_2\). The ideas here can be extended to more variables, and the resulting distribution is called the “multivariate normal”. The bivariate normal is a special case of the multivariate normal.
The bivariate normal distribution has 5 parameters: two means (for \(X_1\) and \(X_2\)), two variances (for \(X_1\) and \(X_2\)) and the covariance between \(X_1\) and \(X_2\). It is usual to write the mean parameter as a vector \(\mu\) and the variance and covariance parameters as a 2x2 symmetric matrix \(\Sigma\), where the diagonal elements of \(\Sigma\) contain the variances and the off-diagonal elements contain the covariance. \(\Sigma\) is called the “covariance matrix” (or sometimes the “variance-covariance matrix”).
Suppose \(Z_1,Z_2\) are independent random variables each with a standard normal distribution \(N(0,1)\). Let \(Z\) denote the vector \((Z_1,Z_2)\), let \(A\) be any \(2 \times 2\) matrix, and \(\mu\) be any \(r\)-vector. Then the vector \(X = AZ+\mu\) has a bivariate normal distribution with mean \(\mu\) and variance-covariance matrix \(\Sigma=AA'\). (Here \(A'\) means the transpose of the matrix \(A\).) We write \(X \sim N_2(\mu,\Sigma)\).
We can redo the example above in vector and matrix notation, with \(\mu=(0,0)\) and \(A=(1,0.1),(1,-0.1)\). Here for clarity we just simulate a single sample instead of 1000:
mu = c(0,0) A = rbind(c(1,0.1),c(1,-0.1)) A
[,1] [,2] [1,] 1 0.1 [2,] 1 -0.1
z = rnorm(2) x = mu + A %*% z x
[,1] [1,] -0.5002188 [2,] -0.7154634
It should be clear from the above that in our example the mean is \(\mu=(0,0)\). What is the covariance matrix \(\Sigma\)? We can compute it from the formula \(\Sigma = AA'\):
Sigma = A %*% t(A) Sigma
[,1] [,2] [1,] 1.01 0.99 [2,] 0.99 1.01
R version 4.1.0 Patched (2021-07-20 r80657) Platform: aarch64-apple-darwin20 (64-bit) Running under: macOS Monterey 12.2 Matrix products: default BLAS: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib locale:  en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8 attached base packages:  stats graphics grDevices utils datasets methods base loaded via a namespace (and not attached):  Rcpp_1.0.7 whisker_0.4 knitr_1.36 magrittr_2.0.2  workflowr_1.7.0 R6_2.5.1 rlang_0.4.12 fastmap_1.1.0  fansi_0.5.0 highr_0.9 stringr_1.4.0 tools_4.1.0  xfun_0.28 utf8_1.2.2 git2r_0.29.0 jquerylib_0.1.4  htmltools_0.5.2 ellipsis_0.3.2 rprojroot_2.0.2 yaml_2.2.1  digest_0.6.28 tibble_3.1.6 lifecycle_1.0.1 crayon_1.4.2  later_1.3.0 vctrs_0.3.8 fs_1.5.0 promises_184.108.40.206  glue_1.5.0 evaluate_0.14 rmarkdown_2.11 stringi_1.7.5  compiler_4.1.0 pillar_1.6.4 httpuv_1.6.3 pkgconfig_2.0.3
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