How to create Non-Central Student's T distribution and what priors to use with the distribution?

Questions
2

I redesigned the custom model to include the custom distribution, however, I keep on getting error based on the equations used to get the likelihood distribution or sometimes tensor locks down and the code just freeze. Find my code below,

with pm.Model() as model:
                # Prior Distributions for unknown model parameters:
                mu = pm.Normal('mu', mu=0, sigma=1)
                sd = pm.HalfNormal('sd', sigma=1)
                nc = pm.HalfNormal('nc', sigma=10)
                nu = pm.HalfNormal('nu', sigma=1)
                
                # Custom distribution:
                # observed_data = pm.DensityDist('observed_data', NonCentralStudentT, observed=data_list)

                # Observed data is from a Likelihood distributions (Likelihood (sampling distribution) of observations):
                observed_data = NonCentralStudentT('observed_data', mu=mu, sd=sd, nc=nc, nu=nu, observed=data_list)
                
                # draw 5000 posterior samples
                trace_S = pm.sample(draws=5000, tune=2000, chains=3, cores=1)
        
                # Obtaining Posterior Predictive Sampling:
                post_pred_S = pm.sample_posterior_predictive(trace_S, samples=3000)
                print(post_pred_S['observed_data'].shape)
                print('\nSummary: ')
                print(pm.stats.summary(data=trace_S))
                print(pm.stats.summary(data=post_pred_S))