Question

Memoryless property of exponen
tial distribution is P(X > a+b | X > a) =
​

Answer 1

Answer:

If X is exponential with parameter λ>0, then X is a memoryless random variable, that is P(X>x+a|X>a)=P(X>x), for a,x≥0. From the point of view of waiting time until arrival of a customer, the memoryless property means that it does not matter how long you have waited so far.

Answer 2

Memoryless property of exponential distribution is

P (X > a + b | X > a) = P (X > x).

Step-by-step explanation:

The following derivation is obtained from the memoryless property for X, and a is a non-negative random variable independent of X, not that it is exponential:

$& P(X > a+b\mid X > a)=\frac{P(X > a+b)}{P(X > a)}$

$& =\frac{\int{P}(X > a+b){{f}_{a}}(a)da}{\int{P}(X > a){{f}_{a}}(a)da}$

$& =\frac{\int{P}(X > b)P(X > a){{f}_{a}}(a)da}{\int{P}(X > a){{f}_{a}}(a)da}$

$& =P(X > x)$

If X is exponential with parameter λ>0, then X is a memoryless random variable, that is P ( X > x + a | X > a) = P (X > x ), for a, x ≥ 0. The memoryless property means that it does not matter how long you have waited so far with respect to the waiting time until arrival of a customer.