# U-quadratic distribution

## Where do you meet this distribution?

- Economic
“Job Insecurity isnâ€™t Always Efficient” by David J. Balan and Dan Hanner

## Shape of Distribution

### Basic Properties

- Two parameters are required.
These parameters are minimum and maximum value of variable respectively.

- Continuous distribution defined on bounded range
- This distribution is always symmetric.

### Probability

- Cumulative distribution function
where

- Probability density function
- How to compute these on Excel.

1 2 3 4 5 6 7 8 9 A B Data Description 2 Value for which you want the distribution 1 Value of parameter A 5 Value of parameter B =12/((A4-A3)^3) Vertical scale =(A3+A4)/2 Mean of the distribution Formula Description (Result) =A5*((A2-A6)^3+(A6-A5)^3)/3 Cumulative distribution function for the terms above =A5*(A2-A6)^2 Probability density function for the terms above

### Quantile

- Inverse of cumulative distribution function
where

- How to compute this on Excel.

1 2 3 4 5 6 7 8

A B Data Description 0.5 Probability associated with the distribution 1 Value of parameter A 5 Value of parameter B =12/((A4-A3)^3) Vertical scale =(A3+A4)/2 Mean of the distribution Formula Description (Result) =(3*A2/A5-(A6-A5)^3)^(1/3)+A6 Inverse of the cumulative distribution function for the terms above

## Characteristics

### Mean – Where is the “center” of the distribution? (Definition)

- Mean of the distribution is given as
- How to compute this on Excel

1 2 3 4 5 A B Data Description 8 Value of parameter A 2 Value of parameter B Formula Description (Result) =(A2+A3)/2 Mean of the distribution for the terms above

### Standard Deviation – How wide does the distribution spread? (Definition)

- Variance of the distribution is given as
Standard Deviation is a positive square root of Variance.

- How to compute this on Excel

1 2 3 4 5

A B Data Description 8 Value of parameter A 2 Value of parameter B Formula Description (Result) =SQRT(3)*(A3-A2)/(2*SQRT(5)) Standard deviation of the distribution for the terms above

### Skewness – Which side is the distribution distorted into? (Definition)

- Skewness of the distribution is .

### Kurtosis – Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

- Kurtosis of the distribution is given as
- How to compute this on Excel

1 2 3 4 5 A B Data Description 8 Value of parameter A 2 Value of parameter B Formula Description (Result) =3*(A3-A2)^4/112 Kurtosis of the distribution for the terms above

## Random Numbers

- Random number x is generated by inverse function method, which is for uniform random U,
where

- How to generate random numbers on Excel.

1 2 3 4 5 6 7

A B Data Description 1 Value of parameter A 5 Value of parameter B =12/((A3-A2)^3) Vertical scale =(A2+A3)/2 Mean of the distribution Formula Description (Result) =(3*NTRAND(100)/A2-(A3-A2)^3)^(1/3)+A3 100 U-quadratic deviates based on Mersenne-Twister algorithm for which the parameters above Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A7:A106 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

## NtRand Functions

Not supported yet