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# 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

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.

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A B
Data Description
0.5 Value for which you want the distribution
1 Value of parameter A
5 Value of parameter B
Formula Description (Result)
=(A2-A3)/(A4-A3) Cumulative distribution function for the terms above
=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.

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A B
Data Description
0.5 Probability associated with the distribution
1 Value of parameter A
5 Value of parameter B
Formula Description (Result)
=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

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

Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

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A B
Data Description
8 Value of parameter A
2 Value of parameter B
Formula Description (Result)
=(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.

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A B
Data Description
1 Value of parameter A
5 Value of parameter B
Formula Description (Result)
=(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