Solving quadnomial distributions

Here we show how the coefficients from the quadnomial probability distribution fit the multinomial formula.  Due to symmetry of a quadnomial vector taken to the fourth power, the number of matrix multiplications is minimized to the square of two identical 4x4 diagonal symmetrical matrices.

We start with the total probability of four variables, which we only need to take to no greater than the same power as the number of variables (ie., four for a quadnomial).

w + x + y + z

=          1

(w + x + y + z)¹

=         w + x + y + z

(w + x + y + z)²

=         |w                          ||w x y z                  |
           |x                           |
           |y                           |
           |z                           |

=       |w² wx wy wz         |

         |xw x² xy xz            |

         |yw yx y² yz            |

         |zw zx zy z²                 |

(w + x + y + z)³

=       |w² wx wy wz         ||w x y z                  |

         |xw x² xy xz            |

         |yw yx y² yz            |

         |zw zx zy z²                 |

We are able to skip here from three trials (e.g., third power), to completely solving for four trials (e.g., fourth power) of the the quadnomial.

(w + x + y + z)⁴

=         [(w + x + y + z)²]²

=       |w² wx wy wz         ||w² wx wy wz         |

         |xw x² xy xz            ||xw x² xy xz            |

         |yw yx y² yz            ||yw yx y² yz            |

         |zw zx zy z²                 ||zw zx zy z²                 |

=       |a₁₁ a₁₂ a₁₃ a₁₄|

         |a₂₁ a₂₂ a₂₃ a₂₄|

         |a₃₁ a₃₂ a₃₃ a₃₄|

         |a₄₁ a₄₂ a₄₃ a₄₄|

As an example, let’s solve for |a₂₁|, which of course also equals |a₁₂| due to diagonal symmetry.

|a₂₁|

=       |w²                            ||                           |

         |xw                            ||xw x² xy xz       |

         |yw                            ||                           |

         |zw                            ||                           |

=       |w³x w²x² w²xy w²xz               |

         |w²x² wx³ wx²y wx²z               |

         |w²xy wx²y wxy² wxyz            |

         |w²xz wx²z wxyz wxz²                  |

Now we collect all the coefficients from all of the |a_bc| matrices, including |a₂₁|.  The coefficients for all of the variable combinations are:

1         w⁴

4         w³x

4         w³y

4         w³z

1          x⁴

4          wx³

4          x³y

4          x³z

1          y⁴

4          wy³

4          xy³

4          y³z

1          z⁴

4          wz³

4          xz³

4          yz³

12         w²xy

12         w²xz

12         w²yz

6          w²x²

6          w²y²

6          w²z²

6          x²y²

6          x²z²

6          y²z²

12         wx²y

12         wx²z

12         x²yz

12         wy²x

12         wy²z

12         xy²z

12         wxz²

12         wyz²

12          xyz²

24         wxyz

256         sum of coefficients

= 4⁴                  or number of variables to the power the vector is taken to (in this case also 4)

We can see from above that there are five classes of variable combinations and the coefficients equal the combination formula for multinomials, or ₄P_#w,#x,#y,#z = 4!/(#w's! #x's! #y's! #z's!):
()⁴         or w⁴, y⁴, y⁴, z⁴ = 4!/(4!) = 1
()³()                               = 4!/(3!) = 4
()²()²                                       = 4!/(2!2!) = 6
()²()()                                      = 4!/(2!1!1!) = 12

()()()() or wxyz                = 4!/(1!1!1!1!) = 24

So these quadnomial coefficients values align to the purple list above of the 35 variable combinations.  And as an easy test, if all variables have a 25% chance of occurring, then the total probability is equal to 256(25%)⁴ = 256/4⁴ = 1.