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exponential random variable (Definition)

$ X$ is a exponential random variable with parameter $ \lambda>0$ if its probability density function is given for $ x>0$ by

$\displaystyle f_X(x) = \lambda e^{-\lambda x}.$    

To denote this, one usually writes $ X\sim Exp(\lambda)$.

For an exponential random variable $ X$:

  1. $ X$ is commonly used to model lifetimes and duration between Poisson events.
  2. The expected value of $ X$ is given by $ E[X] = \frac{1}{\lambda}$
  3. The variance of $ X$ is given by $ Var[X] = \frac{1}{\lambda^2}$
  4. The moments of $ X$ are given by $ M_X(t) = \frac{\lambda}{\lambda - t}$
  5. It is interesting to note that $ X$ is a gamma random variable with an $ \alpha$ parameter of 1.



"exponential random variable" is owned by mathcam. [ full author list (3) | owner history (2) ]
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Other names:  exponential distribution
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Cross-references: gamma random variable, moments, variance, expected value, events, probability density function, parameter
There are 10 references to this entry.

This is version 4 of exponential random variable, born on 2001-10-26, modified 2004-04-09.
Object id is 528, canonical name is ExponentialRandomVariable.
Accessed 16829 times total.

Classification:
AMS MSC: 62E15 (Statistics :: Distribution theory :: Exact distribution theory)
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