What is the difference between Poisson Distribution and Exponential Distribution?

The Poisson probability distribution often provides a good model for the probability distribution of the number of #Y# "rare" events that occur in space, time, volume, or any other dimension.

Examples include car/industrial accidents, telephone calls handled by a switchboard in a time interval, number of radioactive particles that decay in a particular time period, etc.

A random variable #Y# is said to have a Poisson probability distribution if and only if

#p(y)=(lambda^y)/(y!)e^(-lambda)" "y=0,1,2,...,lambda>0#

Where #lambda# is the average value of #Y#.

The exponential probability distribution is actually a specific case of the gamma probability distribution.

The gamma density function does a sufficient job of modeling the populations associated with random variables that are always nonnegative and yield distributions of data that are skewed (non symmetric) to the right.

A random variable #Y# is said to have a gamma distribution with parameters #alpha>0# and #beta>0# if and only if the density function of #Y# is

#f(y)=(y^(alpha-1)e^(-y)/(beta))/(beta^(alpha)Gamma(alpha))#

and zero elsewhere, where #Gamma(alpha)# is the gamma function with argument #alpha#.

#Gamma(alpha)=int_0^(oo)y^(alpha-1)e^(-y)dy#

A random variable #Y# is said to have an exponential distribution with parameter #beta>0# if and only if the density function of #Y# is

#f(y)=1/(beta)e^(-y/beta)," "0 <= y < oo#

and zero elsewhere.

Essentially, the exponential distribution is the gamma distribution, just with #alpha=1# and #beta=beta#. Note that #Gamma(1)=0! =1#.