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.

	A	B
1	Data	Description
2	0.5	Value for which you want the distribution
3	8	Value of parameter Beta
4	Formula	Description (Result)
5	=1-EXP(-A2/A3)	Cumulative distribution function for the terms above
6	=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.

	A	B
1	Data	Description
2	0.5	Probability associated with the distribution
3	1.7	Value of parameter Beta
4	Formula	Description (Result)
5	=-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.

	A	B
1	Data	Description
2	0.5	Value of parameter Beta
3	Formula	Description (Result)
4	=-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