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

 

[blue_raver22]

for your question. I can understand why you may be confused about the concept of memory in distributions. Let me explain further.

First, let's define what we mean by "memory" in this context. In probability theory, memory refers to the property that the future behavior of a random variable is independent of its past behavior. In other words, the past values of the random variable do not affect its future values.

The exponential distribution is considered memoryless because it follows the property of memory. This means that if we pick a random number from an exponential distribution, the probability of picking a number greater than a certain value is the same, regardless of any previous numbers that were chosen. This is because the exponential distribution has a constant hazard rate, meaning that the probability of an event occurring in a given time interval is the same, regardless of how much time has passed.

On the other hand, the normal distribution does not have a constant hazard rate. Its probability density function is bell-shaped, with a peak at the mean and decreasing as we move away from the mean. This means that the probability of picking a number greater than a certain value is affected by the previous numbers that were chosen. For example, if we have previously chosen a number that is close to the mean, the probability of picking a number greater than a certain value is lower compared to if we had previously chosen a number that is far from the mean.

So, to answer your question, other distributions such as the normal distribution are not considered memoryless because their future behavior is affected by their past behavior. The exponential distribution, on the other hand, is considered memoryless because its future behavior is independent of its past behavior.

I hope this explanation helps clarify the concept of memory in distributions. If you have any further questions, please don't hesitate to ask.