Update basic_distributions syntax

[andrjohns]

I'm going to start (slowly & sporadically) updating the syntax in these example models to modern Stan language, and remove deprecated syntax.

This PR updates the models in the basic_distributions folder. I've checked that all models parse under Stanc3 (with the exception of a known issue

[rok-cesnovar]

Nice! Thanks!

Do you check using rstan 2.26? cmdstan adds a _model suffix and rstan should as well to avoid the problem.

[andrjohns]

Ah good call, yep no errors under RStan 2.26

[rok-cesnovar]

The auto-formatter from stanc3 suggests a bit different formatting in some places, though I guess its a matter of personal taste a bit so feel free to ignore.

Thanks for doing this!

[rok-cesnovar]

Suggested change

	matrix[2,2] Sigma = [[1, 0.1],
	[0.1, 1]];
	matrix[2,2] Sigma = [[1, 0.1], [0.1, 1]];

[rok-cesnovar]

Suggested change

	cov_matrix[3] S = [[2, 0, 0],
	[0, 1, 0],
	[0, 0, 0.5]];
	cov_matrix[3] S = [[2, 0, 0], [0, 1, 0], [0, 0, 0.5]];

[rok-cesnovar]

Suggested change

	cov_matrix[2] S = [[2, 0],
	[0, 0.5]];
	cov_matrix[2] S = [[2, 0], [0, 0.5]];

[rok-cesnovar]

If the above was filling the lower triangular parts we could use symmetrize_from_lower_tri but no sense complicating I guess.

[rok-cesnovar]

Suggested change

	cov_matrix[2] S = [[2, 0],
	[0, 0.5]];
	cov_matrix[2] S = [[2, 0], [0, 0.5]];

[rok-cesnovar]

Suggested change

	matrix[2,2] W = [[sd1^2, cov],
	[cov, sd2^2]];
	matrix[2,2] W = [[sd1 ^ 2, cov], [cov, sd2 ^ 2]];

[andrjohns]

Just for the example models I think I'd prefer to leave the matrix declarations in the matrix layout (rather than a one-liner), so that the [[],[]] declaration syntax is a bit more immediately intuitive for the unfamiliar. But will update the rest

[rok-cesnovar]

Yeah, its why I said these are optional. Let me know when to take another look.

[andrjohns]

Works for me! Ready for another look

[rok-cesnovar]

Thanks!