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# Exponential distribution

## Where do you meet this distribution?

• The lengths of the inter-arrival times in a homogeneous Poisson process
• Nuclear physics : The time until a radioactive particle decays
• Statistical mechanics : Molecular distribution in uniform gravitational field
• Risk management : The time until default in reduced form credit risk modeling

## Shape of Distribution

### Basic Properties

• A parameter $\beta$ is required.
$\beta>0$

This parameter is Mean of the distribution.

• Continuous distribution defined on semi-infinite range $x \geq 0$
• This distribution is always asymmetric.

### Probability

• Cumulative distribution function
$F(x)=1-\exp\left(-\frac{x}{\beta}\right)$
• Probability density function
$f(x)=\frac{1}{\beta}\exp\left(-\frac{x}{\beta}\right)$
• How to compute these on Excel.

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Data Description
0.5 Value for which you want the distribution
8 Value of parameter Beta
Formula Description (Result)
=1-EXP(-A2/A3) Cumulative distribution function for the terms above
=EXP(-A2/A3)/A3 Probability density function for the terms above

### Quantile

• Inverse function of cumulative distribution function
$F^{-1}(P)=-\beta\ln(1-P)$
• How to compute this on Excel.

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Data Description
0.5 Probability associated with the distribution
1.7 Value of parameter Beta
Formula Description (Result)
=-A3*LN(1-A2) 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 $\beta$.

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

• Standard deviation of the distribution is given as $\beta$.

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

• Skewness is $2$.

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

• Kurtosis is $6$.

## Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,
$x=-\beta\ln(1-U)$
• How to generate random numbers on Excel.

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Data Description
0.5 Value of parameter Beta
Formula Description (Result)
=-A2*LN(1-NTRAND(100)) 100 exponential 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 A4:A103 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

## NtRand Functions

Not supported yet