Exponential distribution, memory

[oneamp]

I am told that an exponential distribution is memoryless. But why aren't other distributions, such as the normal distribution, also memoryless? If I pick a random number from an exponential distribution, it is not effected by previously chosen random numbers. But isn't that also the case for a normal distribution, for example?

What do I misunderstand?

Thank you

 

[economicsnerd]

I think when people say the exponential distribution is memoryless, they mean that, for any exponentially distributed random variable $X$, the distribution of $X-x$ conditional on the event $\{X\geq x\}$ is the same as the distribution of $X$.

It's easy to check that that the normal distribution (or, in fact, anything that can take on negative values) can't satisfy the above property.

 

[statdad]

The memory less notion refers to the fact that for any positive numbers s and t

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

It can be shown that this property gives the exponential distribution as the unique continuous distribution with the property.

 

[oneamp]

Thank you