Why uniform distribution is not memoryless?

Question

The exponential and geometric distributions have the memoryless property, meaning that the distribution of the waiting times between the events does not depend on how much time has elapsed already. But I'm trying to intuitively understand why uniform distribution is not memoryless. Can someone please help me with that?

Maybe this example will explain what is my concern:

Scenario 1: We have a room, to which $k$ identical people arrived (the arrived at different times). Each person stayed in the room a random amount of time $x$, where $x$ is from exponential distribution. Now, I observe one person leaving - the probability that this person is the same one who entered the room first, is the same as the probability it was the second one, the third one etc. So, the person leaving the room can be with equal chances any of the $k$ people.

Scenario 2: I have the same story, but now the people do not wait random exponential time. Instead, when the people enter the room one person is picked uniformly at random to leave the room. Then the next one, and the next one.

So, given the uniform distribution is not memoryless, in the second scenario can I somehow tell which of the incoming persons is no leaving? If not, how is this different from the memoryless property?

Answer 1

Scenario 2 has nothing to do with the memoryless property. Saying that you pick one person at random is the same as scenario 1, since in that scenario, due to the memoryless of the exponential distribution, any person has the same probability to finish first.

To change the story: in scenario 2, each person has to wait $U(0,T)$ before leaving. If the first person entered at $t=0$ and the second person at $T-\epsilon$ without the first leaving, there is a probability of $1$ that #1 will leave in the next $\epsilon$ seconds while only $\epsilon/T$ that #2 will. Hence the memorylessness: the waiting time urged #1 to leave soon. If the waiting time was exponential, no matter how much we waited, both could have left with the same probability.

Answer 2

First the definition of memoryless property. Let $X$ be a random variable, it has the memoryless property if for any $a$, $b$

$$P(X>a+b|X>a)=P(X>b)$$

What does this mean? It means that for example, if $X$ is a waiting time, if we have waited for $a$ minutes, the probability we are let in, in the next $b$ minutes is the same, regardless of how much we waited.

And why does uniform distribution not have this property? One way to understand is that, uniform distribution has an "end", therefore we know for certain that we are let in, in less and less minutes the more we wait. If we have waited the entire "length" of the distribution we are let in immediately for certain. Consider $U(a,b)$, then mathematically

$$P(U>a+b|U>b-\epsilon)\approx0 \neq 1=P(U>a)$$

As for the property that is outlined in scenario 1 and 2, this has nothing to do with memory. What you are saying is that the people are independently picked, without order. To have this property we can use any independent random variables, summing them for the different waiting times. E.g. $X_1$ is the waiting time of the first person $X_1+X_2$ is the waiting time of the second, where $X_1,X_2,...$ are $IID$. Further we have a other discrete uniform $RV$ for who gets picked. Why is it not memoryless? Because for each person the more they wait, the odds of them being picked can change.