How to Incorporate prior information on CDF values into the model?

Modeling
1

I am working with betting data where the closing odds can be thought of the median result.

I am trying to estimate the latent offensive and defensive strength of each team. Thus, I am using the observed scores and I would also like to use the closing odds to incorporate extra information into my model.

  data {
    // pbserved scores
    vector[30] home_scores; 
    vector[30] away_scores; 
    
    int num_teams;
    
    int home_team_index[30];
    int away_team_index[30];
    
    
    // this is the markets' median value of home team score - away team score 
    vector[30] closing_spread;
  }
  
  parameters{
    
    vector[num_teams] offensive_strength;
    vector[num_teams] defensive_strength;
    
    vector[num_teams] off_sigma;

  }
  
  model{
    int h;
    int a; 
    
    real home_mu[30];
    real away_mu[30];
    
    vector[30] home_sigma;
    vector[30] away_sigma;
    
    vector[30] probability_beat_spread;
    
    offensive_strength ~ normal(30,10);
    defensive_strength ~ normal(0,10);
    
    
    for(i in 1:30){
      h = home_team_index[i];
      a = away_team_index[i];
      home_mu[i] = offensive_strength[h] + defensive_strength[a];
      away_mu[i] = offensive_strength[a] + defensive_strength[h];
      home_sigma[i] = off_sigma[h];
      away_sigma[i] = off_sigma[a];
      
      
      probability_beat_spread[i] = normal_cdf(closing_spread[i], home_mu[i] - away_mu[i], sqrt(square(home_sigma[i])+ square(away_sigma[i])));
          
      }
       
    home_scores ~ normal(home_mu, home_sigma);
    away_scores ~ normal(away_mu, away_sigma);
    
    // here I want to specify that probability_beat_spread should somewhere around .5 
    // I know I can do something like :
    probability_beat_spread ~ normal(.5,.05); 
    
    // this significantly changes my estimates however, and gives me really wierd results so I don't think this is correct
    
    // How do I incorporate this prior information about the spread being the median point
    }
2

I’m afraid I don’t have enough information to start helping you. Can you share a bit more of your model? Mathematically, if possible. It’s unclear to me what you need.