# Poisson Distribution

## From June 01, 2020

The Poisson distribution represents the probability of the distribution of a number of events occurred in fixed interval of time, if this events happen independently and with a fixed constant mean defined by `λ`

.

The values of that random variable `X = 0, 1, 2, .... ,`

and its probability's distribution can be evaluated by

The distribution's law can also be found by the following:

Where `t`

is the length of the observation's interval, `λ`

is the intensity of the occurrences of the event by unit of time.

So given the following *generator*, the simulation's algorithm is the following:

## Result

### Team

```
class Team
{
public int ID { get; private set; }
public bool Flag { get; set; }
public int AverageGoals { get; private set; }
private Random rand;
public Team(int ID)
{
this.ID = ID;
this.Flag = false;
rand = new Random(ID);
this.AverageGoals = rand.Next(8);
}
```

### Helpers

```
private const int ELIMINATED = -1;
private float Poisson(int miu, int x)
{
int factorial = 1;
for(int i = x; i > 0; i--)
{
factorial *= i;
}
return (float) (Math.Pow(miu, x) * (Math.Exp(-miu)) / factorial);
}
private Team GetWinner(Team t1, Team t2)
{
Random rand = new Random();
int goalsTeam1 = rand.Next(10);
int goalsTeam2 = rand.Next(10);
float n1 = Poisson(t1.AverageGoals, goalsTeam1);
float n2 = Poisson(t2.AverageGoals, goalsTeam2);
return n1 > n2 ? t1 : t2;
}
```

### Main

```
int rounds = 0;
int num_teams = 0;
num_teams = int.Parse(textBox1.Text);
rounds = (int) Math.Log(num_teams, 2.0);
List<Team> teams = new List<Team>();
for (int i = 0; i < num_teams; i++)
{
Team temp = new Team(i + 1);
teams.Add(temp);
}
// Display the result.
string txt = "";
for (int round = 0; round < rounds; round++)
{
int n1 = teams.Count;
txt += "Round " + (round + 1) + ":\r\n";
for (int i = 0; i < (n1 * 0.5); i++)
{
Team team1 = teams[i];
Team team2 = teams[n1 - (i + 1)];
Team winner = GetWinner(team1, team2);
txt += " Team" + (team1.ID) + " VS Team " + (team2.ID) + " -> " + (winner.ID) + "\r\n";
if (winner == team1)
{
teams[i].Flag = true;
teams[n1 - (i + 1)].Flag = false;
//txt += " Team" + (team2.ID) + " (ELIMINATED)\r\n";
}
else
{
teams[i].Flag = false;
teams[n1 - (i + 1)].Flag = true;
//txt += " Team" + (team1.ID) + " (ELIMINATED)\r\n";
}
}
teams.RemoveAll(x => x.Flag == false);
if(teams.Count == 1)
{
break;
}
}
textBox2.Text = txt;
```