# Exponential distribution, memory

1. Feb 5, 2014

### 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

2. Feb 5, 2014

### 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.

3. Feb 5, 2014

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.

4. Feb 5, 2014

Thank you