Begin2.DVI

The Exponential Distribution

The exponential probability distribution is a continuous probability distribution

with parameter λ > 0 and is defined

f(x) =

{

λ e−λx , for x > 0

0 , otherwise

(11 .72)

The exponential distribution is used in studying time to failure of a piece of equip-

ment , waiting time for next event to occur, like waiting time for an elevator, or

time waiting in line to be served. This distribution has the mean

μ=λ

and the variance is given by

σ^2 =λ^2

Note that the area under the probability curve f(x), for −∞ < x < ∞is equal to 1 or

∫∞

−∞

f(x)dx =

∫∞

0

λe−λx dx = 1

The Gamma Distribution

The gamma probability distribution is defined

f(x) =







1

θαΓ(α)

xα−^1 e−x/θ, for x > 0

0 , for x≤ 0

(11 .73)

where Γ(α) is the gamma function. This probability density function has the two

parameters α > 0 and θ > 0. It is a continuous probability distribution with a shape

parameter αand scale parameter θ. The gamma distribution is used frequently in

econometrics.

This probability distribution arises in determining the waiting time for a given

number of events to occur. For example, waiting for 10 calls to a switch board, or

life testing until a failure occurs. It also occurs in weather prediction of precipitation

processes. The gamma distribution has mean

μ=αθ and variance σ^2 =α θ^2

The gamma distribution with parameters α= 1 and θ= 1/λ produces the exponential

distribution. The figure 11-12 illustrates the gamma distribution for selected values

of the parameters αand θ.