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Let $X_1$, $X_2$, $X_3$ be three random variables with common pairwise correlation coefficient $\rho$, that is $\mbox{corr}(X_i, X_j)= \rho$ for $i \neq j$ with $|\rho|\leq 1$. … So, the correlation matrix of $X = (X_1, X_2, X_3)$ is $$ \left( \begin{array}{ccc} 1 & \rho & \rho \\ \rho & 1 & \rho \\ \rho & \rho & 1 \end{array} \right) . $$ Correlation matrices need to be …

$\rho_{\max}$) denote the lower (resp. upper) bound of the attainable correlation between $X_1$ and $X_2$. … Now, let's compute the correlation of $X_1$ and $X_2$. …

What is the range of attainable correlations for the pair of exponentially distributed random variables $X_1 \sim {\rm Exp}(\lambda_1)$ and $X_2 \sim {\rm Exp}(\lambda_2)$, where $\lambda_1, \lambda_2 …

Attainable correlation for lognormal random variables To obtain $\rho_{\max}$ we use the fact that the maximum correlation is attained if and only if $X_1$ and $X_2$ are comonotonic. … upper bound", "Correlation lower bound")) abline(h = 0, lty = 2) …

Compute ${\rm corr} (I_k, I_j)$ to see the information that is provided in the correlation matrix. …

# Target (Spearman) correlation n <- 500 # Number of samples ## Functions gen.gauss.cop <- function(r, n){ rho <- 2 * sin(r * pi/6) # Pearson correlation P <- … Simulation study The following simulation study repeated for target correlation $r= -0.5, 0.1, 0.6$ suggests that the distribution of the correlation coefficient converges to the desired correlation as …

The variance of $\hat{B}_1$ is $$ {\rm var}(\hat{B}_1) = \sigma^2 ([X^{'}X]^{-1})_{22}, $$ where $\sigma^2 = {\rm var}(e)$, and $X = [\boldsymbol{1}, X_1, X_2]$ denotes the design matrix (see for exam …

The distribution of the sum of independent random variables/vectors can often be obtained easily using moment generating functions (MGF). In short, the MGF of the sum is the product of the MGFs of th …