Exponential Distribution
It is said that “Exponential variables lose memory.”
Suppose that an Exponential variable 𝑇 represents waiting time. Memoryless property means that the fact of having waited for 𝑡 minutes gets “forgotten,” and it does not affect the future waiting time.
Regardless of the event 𝑇 > 𝑡, when the total waiting time exceeds 𝑡, the remaining waiting time still has Exponential distribution with the same parameter. Mathematically,
𝑷 {𝑇 > 𝑡+𝑥 | 𝑇 > 𝑡} = 𝑷 {𝑇 > 𝑥} for 𝑡, 𝑥 > 0.
Proof
given exponential function 𝑷 {𝑇 > 𝑖} = 𝑒-λ𝑖
- 𝑷 {𝑇 > 𝑡+𝑥 | 𝑇 > 𝑡} = 𝑷 {𝑇 > 𝑡+𝑥, 𝑇 > 𝑡} / 𝑷 {𝑇 > 𝑡}
- 𝑷 {𝑇 > 𝑡+𝑥 | 𝑇 > 𝑡} = 𝑷 {𝑇 > 𝑡+𝑥} / 𝑷 {𝑇 > 𝑡}
- 𝑷 {𝑇 > 𝑡+𝑥 | 𝑇 > 𝑡} = 𝑒-λ(𝑡+𝑥) / 𝑒-λ𝑡
- 𝑷 {𝑇 > 𝑡+𝑥 | 𝑇 > 𝑡} = 𝑒-λ𝑥𝑒-λ𝑡/ 𝑒-λ𝑡
- 𝑷 {𝑇 > 𝑡+𝑥 | 𝑇 > 𝑡} = 𝑒-λ𝑥
- 𝑷 {𝑇 > 𝑡+𝑥 | 𝑇 > 𝑡} = 𝑷 {𝑇 > 𝑥}
Geometric Distribution
geometric distribution also has this memoryless property