The Normal distribution is a continuous probability distribution with two parameters:

• the mean ("average"), usually written \(\mu\) (mu)
• the standard deviation ("variability"), usually written \({\sigma}\) (sigma). (Note: standard deviation is \(\sigma\), variance is \({\sigma}^2\)).

The standard normal distribution is the normal distribution with a mean of zero and a variance of one.

Stan provides the function normal_rng which takes two real-values arguments mu and sigma, (note: sigma must be positive). This function may only be used in generated quantities block.

The shape of the probability densities functions for these distributions are all bell-shaped, however they have more area under the tail of the curve than the Normal does, thus the extreme quantiles are farther from the mean. For the Cauchy distribution, the area under the tail is infinite! These distributions are useful for random variables whose variance is unknown.

Stan provides functions cauchy_rng, student_t_rng, and logistic_rng, all of which take location and scale parameters mu and sigma.

The multivariate normal distribution is the extension of the normal distribution of a single random variable to a vector of random variables (a "random vector"). Each element of a random vector is a variable which is normally distributed, i.e., each element is drawn from its own normal distribution with parameters mu and sigma.

Generating a standard normal random vector in Stan is just like generating a standard normal variate, only using a vector and a for loop:

vector[N] rv;
for (i in 1:N) rv[i] = normal_rng(0, 1);

• What is the Expectation of a random variable generated from a standard normal distribution?

• What is the Expectation of a random vector generated from a standard multinormal distribution?

Program gen_norm.stan: generate a sample for a random variate, location = 1, scale = 3:

generated quantities {
  real rv = normal_rng(1,3);
}

Generate a sample consisting of 4000 draws, extract generated quantity rv:

f1 = stan("gen_norm.stan", algorithm = "Fixed_param");
xs = as.data.frame(f1, pars=("rv"))

Create a histogram, overlay with density curve:

pp = ggplot(data=xs, aes(xs$rv));
pp = pp + geom_histogram(aes(y = ..density..), bins=50, fill = "darkgrey");
pp = pp + stat_function(fun = dnorm,
          args = list(mean = mean(xs$rv), sd = sd(xs$rv)), lwd=2, col="red");

• Write a program gen_norm_variate.stan which generates a random variate x for some choice of mu and sigma. You can either specify mu and sigma directly in the program or pass these values in as data.

• Use this program to generate a sample, as before, then create a histogram plot to show the resulting density of x when mu is 0 and sigma is 1.

• Repeat this several times, choosing both positive and negative values of mu, and values for sigma less than 1 as well as greater than 1.

• Write a program gen_cauchy_variate.stan which generates a random variate x for some choice of mu and sigma.

• Use this program to generate a sample

• Using the same values of mu and sigma, generate a sample using program gen_norm_variate.stan

• Create histogram plots for both, compare and contrast.

• Use the ggplot magic to overlay the normal density curve, discuss.

• Write a program gen_std_norm_vector.stan which generates a standard normal random vector X of length N for some choice of N and computes its distance to the origin vector as quantity of interest dist_to_origin. Generalize this program to compute distance to the origin for all standard normal random vectors of length 1 through N.

• Generate an x,y plot for x in 1:256 where x is the length of the random vector X and y is the expectation of the distance to the origin when X is a standard normal random vector.