# 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 is required.
This parameter is Mean of the distribution.

- Continuous distribution defined on semi-infinite range
- This distribution is always asymmetric.

### Probability

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

1 2 3 4 5 6 A B 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
- How to compute this on Excel.

1 2 3 4 5 A B 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 .

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

- Standard deviation of the distribution is given as .

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

- Skewness is .

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

- Kurtosis is .

## Random Numbers

- Random number x is generated by inverse function method, which is for uniform random U,
- How to generate random numbers on Excel.

1 2 3 4

A B 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

## Reference

- Wolfram Mathworld – Exponential distribution
- Wikipedia – Exponential distribution
- Statistics Online Computational Resource