GEOMETRIC AND EXPONENTIAL
Analysis of the relationship between the two random variables
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A geometric random variable is related to the exponential random variable. Here is the derivation:
If an interval of length T is divided into subintervals of length
The sequence of subintervals corresponds to a sequence of Bernoulli trials with probability of success , where a is the average number of events per T seconds.
The number of subintervals until an event is a geometric random variable M.
The time until the occurance of the first event is and the probability that this exceeds t seconds is
Applying the geometric probability pmf
As the size of the bins is decreased and thus, the number of bins is increased
If we definethen so,
which is the pdf for an exponential random variable.
This relationship is used in the relationship between Poisson & exponential random variables.
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RELATED
CONCEPTSRandom Variables Probability Density Function Probability Mass Function Cumulative Distribution Function Expectation of a Discrete Random Variable Important Discrete Random Variables ![]()
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