Uniform Distribution (Continuous)

Where will you meet this distribution?

• Generating random numbers according to a desired distribution
• Digital signal processing (dithering) -- digital audio, digital video, digital photography, seismology, RADAR, weather forecasting systems and many more

Shape of Distribution

Basic Properties

• Two parameters $a, b$ are required.

  $$a<b$$

  These parameters are minimum and maximum value of variable respectively.

• Continuous distribution defined on bounded range $a\leq x \leq b$

• This distribution is always symmetric.

Probability

• Cumulative distribution function

  $$F(x)=\frac{x}{b-a}$$

• Probability density function

  $$f(x)=\frac{1}{b-a}$$

• How to compute these on Excel.

	A	B
1	Data	Description
2	0.5	Value for which you want the distribution
3	1	Value of parameter A
4	5	Value of parameter B
5	Formula	Description (Result)
6	=(A2-A3)/(A4-A3)	Cumulative distribution function for the terms above
7	=1/(A4-A3)	Probability density function for the terms above

Quantile

• Inverse of cumulative distribution function

  $$F^{-1}(P)=a+P(b-a)$$

• How to compute this on Excel.

	A	B
1	Data	Description
2	0.5	Probability associated with the distribution
3	1	Value of parameter A
4	5	Value of parameter B
5	Formula	Description (Result)
6	=A3+A2*(A4-A3)	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

  $$\frac{a+b}{2}$$

• How to compute this on Excel

	A	B
1	Data	Description
2	8	Value of parameter A
3	2	Value of parameter B
4	Formula	Description (Result)
5	=(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

  $$\frac{(a-b)^2}{12}$$

  Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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

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

• Skewness is $0$

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

• Kurtosis is $-1.2$

Random Numbers

• Random number x is generated from uniform random U,

  $$x=a+U(b-a)$$

• How to generate random numbers on Excel.

	A	B
1	Data	Description
2	8	Value of parameter A
3	5	Value of parameter B
4	Formula	Description (Result)
5	=(A3-A2)*NTRAND(100,A2,A3,0)+A2	100 uniform 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 A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

• Generating random numbers based on Mersenne Twister algorithm: NTRAND

Reference

• Wolfram Mathworld -- Uniform Distribution
• Wikipedia -- Uniform distribution (continuous)
• Statistics Online Computational Resource