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- Function:
*int***gsl_ran_multivariate_gaussian***(const gsl_rng **`r`, const gsl_vector *`mu`, const gsl_matrix *`L`, gsl_vector *`result`) -
This function generates a random vector satisfying the

*k*-dimensional multivariate Gaussian distribution with mean*\mu*and variance-covariance matrix*\Sigma*. On input, the*k*-vector*\mu*is given in`mu`, and the Cholesky factor of the*k*-by-*k*matrix*\Sigma = L L^T*is given in the lower triangle of`L`, as output from`gsl_linalg_cholesky_decomp`

. The random vector is stored in`result`on output. The probability distribution for multivariate Gaussian random variates isp(x_1,...,x_k) dx_1 ... dx_k = {1 \over \sqrt{(2 \pi)^k |\Sigma|} \exp \left(-{1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu)\right) dx_1 \dots dx_k

- Function:
*int***gsl_ran_multivariate_gaussian_pdf***(const gsl_vector **`x`, const gsl_vector *`mu`, const gsl_matrix *`L`, double *`result`, gsl_vector *`work`) - Function:
*int***gsl_ran_multivariate_gaussian_log_pdf***(const gsl_vector **`x`, const gsl_vector *`mu`, const gsl_matrix *`L`, double *`result`, gsl_vector *`work`) These functions compute

*p(x)*or*\log{p(x)}*at the point`x`, using mean vector`mu`and variance-covariance matrix specified by its Cholesky factor`L`using the formula above. Additional workspace of length*k*is required in`work`.

- Function:
*int***gsl_ran_multivariate_gaussian_mean***(const gsl_matrix **`X`, gsl_vector *`mu_hat`) Given a set of

*n*samples*X_j*from a*k*-dimensional multivariate Gaussian distribution, this function computes the maximum likelihood estimate of the mean of the distribution, given by\Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j

The samples

*X_1,X_2,\dots,X_n*are given in the*n*-by-*k*matrix`X`, and the maximum likelihood estimate of the mean is stored in`mu_hat`on output.

- Function:
*int***gsl_ran_multivariate_gaussian_vcov***(const gsl_matrix **`X`, gsl_matrix *`sigma_hat`) Given a set of

*n*samples*X_j*from a*k*-dimensional multivariate Gaussian distribution, this function computes the maximum likelihood estimate of the variance-covariance matrix of the distribution, given by\Hat{\Sigma} = {1 \over n} \sum_{j=1}^n \left( X_j - \Hat{\mu} \right) \left( X_j - \Hat{\mu} \right)^T

The samples

*X_1,X_2,\dots,X_n*are given in the*n*-by-*k*matrix`X`and the maximum likelihood estimate of the variance-covariance matrix is stored in`sigma_hat`on output.