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Student[Statistics][ExponentialRandomVariable] - exponential random variable

Calling Sequence

ExponentialRandomVariable(b)

Parameters

b

-

scale parameter

Description

• 

The exponential random variable is a continuous probability random variable with probability density function given by:

ft&equals;piecewiset<0&comma;0&comma;&ExponentialE;tbb

  

subject to the following conditions:

0<b

• 

The exponential random variable has the lack of memory property: the probability of an event occurring in the next time interval of an exponential distribution is independent of the amount of time that has already passed.

• 

The exponential variate with scale parameter b is a special case of the Gamma variate with scale parameter b and shape parameter 1: Exponential(b) ~ Gamma(b,1)

• 

The exponential variate with scale parameter b is a special case of the Weibull variate with scale parameter b and shape parameter 1: Exponential(b) ~ Weibull(b,1)

• 

The exponential variate with scale parameter b is related to the unit Uniform variate by the formula:  Exponential(b) ~ -b * log(Uniform(0,1))

• 

The discrete analog of the exponential variate is the Geometric variate.

• 

The exponential variate with scale parameter b is related to the Laplace variate with location parameter a and scale parameter b according to the formula:  Exponential(b) ~ abs(Laplace(a,b) - a).

Examples

withStudent&lsqb;Statistics&rsqb;&colon;

X:=ExponentialRandomVariableb&colon;

PDFX&comma;u

&lcub;0u<0&ExponentialE;ubbotherwise

(1)

PDFX&comma;0.5

&ExponentialE;0.5bb

(2)

MeanX

b

(3)

VarianceX

b2

(4)

Y:=ExponentialRandomVariable3&colon;

PDFY&comma;x&comma;output&equals;plot

CDFY&comma;x

&lcub;0x<01&ExponentialE;13xotherwise

(5)

CDFY&comma;4&comma;output&equals;plot

See Also

Statistics[Distributions][Exponential], Student, Student[Statistics], Student[Statistics][RandomVariable]

References

  

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

  

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

  

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.


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