Show that an exponential density has memoryless property Exp

Show that an exponential density has memoryless property. (Exponential density is the only density function for which the memoryless property holds. Recall that, among discrete distributions, geometric pdf is the only one for which the property holds)

Solution

We have given the exponential density and we want to show that exponential density has memoryless property.

X~exp() , X>0 and >0

A variable X with positive support is memoryless if for all t > 0 and s > 0,

P(X > s + t | X > t) = P(X > s)

using the definition of conditional probability,

P(X > s + t) = P(X > s)P(X > t).

Consider the CDF of X is ,

F(a) = P(X a) = 1 - e^-a

Consider LHS = P(X >s+t and X>t) / P(X>t)

= P(X>s+t) * P(X>t) / P(X>t)

= P(X>s+t) ____________*)

Now P(X>s+t) = 1 - P( X t ) = 1 - (1-e^-(s+t) )

  P(X>s+t)  = e^-(s+t) _______1)

and P(X>t) = e^-t _______2)

divide equation 1) byequation 2),

P(X>s+t) / P(X>t) = e^-(s+t) / e^-t

= e^-s _____3)

RHS = P(X>s) =  e^-s _____4)

We see that equation 3) and 4) are equal.