عجفت الغور

probability

Tags: statistics

Probability Simplex

  • A shape where each vertex represents a probability
  • Example:
Outcomes: {1, 2, 3}
Let the probabilities be p1​,p2​,p3​.
We know that p1​≥0, p2​≥0, p3​≥0, and p1​+p2​+p3​=1.
Geometrically, this can be represented by a 2-simplex, which is an equilateral triangle.

Each vertex of the triangle represents a distribution where one outcome has a probability of 1 and the others have a probability of 0.
        Vertex 1: p1​=1,p2​=0,p3​=0
        Vertex 2: p1​=0,p2​=1,p3​=0
        Vertex 3: p1​=0,p2​=0,p3​=1
    Any point inside the triangle (or on its edges) represents a probability distribution where p1​,p2​, and p3​ are non-negative and sum to 1. The closer a point is to a vertex, the higher the probability of that corresponding outcome.