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February 24, 2021
• The Probability Density Function(PDF) of the exponential random variable t is given
by:
f(t) = λe−λt, t ≥ 0
• Mean= 1/λ
• The Cumulative Distribution Function(CDF) :
F(t) = f(T ≤ t) = 1 − e
−λt t ≥ 0
• Inverse transform sampling is a basic method for generating sample numbers at random
from any probability distribution given its cumulative distribution function. This means by
knowing the value of F(t) we can find the value of t as follows:
t = −
1
λ
ln(1 − F(t))
In Jave, the value of F(t) can be generated using nextDouble() function which returns the
next pseudorandom, uniformly distributed value between 0 and 1 from this random number
generator’s sequence.
• Example
Generate an arrival time from an exponential distribution of mean 3, if nextDouble()
returns 0.811. Sol. the arrival time value t can be found by substituting t=0.811 in the
above equation:
t = −
1
λ
ln(1 − F(t)) = −
1
3
ln(1 − 0.811) = 5
1
February 24, 2021
• The Probability Density Function(PDF) of the exponential random variable t is given
by:
f(t) = λe−λt, t ≥ 0
• Mean= 1/λ
• The Cumulative Distribution Function(CDF) :
F(t) = f(T ≤ t) = 1 − e
−λt t ≥ 0
• Inverse transform sampling is a basic method for generating sample numbers at random
from any probability distribution given its cumulative distribution function. This means by
knowing the value of F(t) we can find the value of t as follows:
t = −
1
λ
ln(1 − F(t))
In Jave, the value of F(t) can be generated using nextDouble() function which returns the
next pseudorandom, uniformly distributed value between 0 and 1 from this random number
generator’s sequence.
• Example
Generate an arrival time from an exponential distribution of mean 3, if nextDouble()
returns 0.811. Sol. the arrival time value t can be found by substituting t=0.811 in the
above equation:
t = −
1
λ
ln(1 − F(t)) = −
1
3
ln(1 − 0.811) = 5
1