Interpretation of the maximum value of a CDF

[dranglerangus]

I was watching a youtube video from MIT's open courseware series on probability. A scenario was proposed: Al is waiting for a bus. The probability that the bus arrives in x minutes is described by the random variable X, which is uniformly distributed on the interval [0,10] (in minutes).

I understand that the cdf of this function is F(x) = {0 for x<0}, {x for 0≤x≤10}, and {1 for x>10}.

This says that the probability is 1 that x≤10, or equivalently that x∈[0,10], right? So does this mean that the bus definitely arrived between 0 and 10 minutes? This seems counter-intuitive, since at any given moment Al was waiting, the probability that the bus would show up was 1/10. To me, this doesn't seem to guarantee that the bus would have to show up at some time in that interval.

The probability laws guarantee that the bus will come within 10 minutes, but it doesn't seem right to me. Am I understanding this incorrectly?

 

[mathman]

The assumption that the bus will definitely arrive in ten minutes is just that - an assumption. For the purpose of doing mathematics, one can assume anything consistent with the rules of probability. It does not have to be physically realistic.

Also you need to distinguish between probability at the start (1/10 for each minute) and conditional probability after he has been waiting a while (after waiting 6 minutes the probability for each of the remaining minutes is now 1/4).

 

[dranglerangus]

The assumption that the bus will definitely arrive in ten minutes is just that - an assumption. For the purpose of doing mathematics, one can assume anything consistent with the rules of probability. It does not have to be physically realistic.

Also you need to distinguish between probability at the start (1/10 for each minute) and conditional probability after he has been waiting a while (after waiting 6 minutes the probability for each of the remaining minutes is now 1/4).

Hmm...weird. It doesn't seem like the probability should change just because he has been waiting a while. How would you formulate that conditional probability?

 

[Stephen Tashi]

{x for 0≤x≤10}

Should be "x/10"

This says that the probability is 1 that x≤10, or equivalently that x∈[0,10], right?

Yes.

So does this mean that the bus definitely arrived between 0 and 10 minutes?

From a practical point of view, yes.

However, in mathematics there is a technical difference between an event having probability 1 and an event being a definite fact. For example, in logic we have the rule modus ponens which says

If A ithen B
A
-----
Therefore B

where A and B are statements.

However there is no rule of loic that says

If A then B
A is true with probability 1
-----
Therefore B

This seems counter-intuitive, since at any given moment Al was waiting, the probability that the bus would show up was 1/10.

That is not a correct interpretation of the probability density function f(x) = 1/10. The value of a pdf at a particular x is not the probability that the value x will actually be realized.

By analogy, if the mass density of a 10 meter rod is 1/10 kg per meter, this does not mean that the weight of the rod "at" x = 3 meters is 1/10 kg.

Many rules of probability can be remembered by thinking of the pdf f(x) as giving "the probability of x", however this occasionally will lead to wrong conclusions. It's analogous to thinking of dy and dx in calculus as "infinitesimal numbers" That's useful, but it isn't an infallible way of reasoning.

 

[dranglerangus]

Should be "x/10"

Whoops!

However, in mathematics there is a technical difference between an event having probability 1 and an event being a definite fact. For example, in logic we have the rule modus ponens which says

If A ithen B
A
-----
Therefore B

where A and B are statements.

However there is no rule of loic that says

If A then B
A is true with probability 1
-----
Therefore B

This is really what I was asking, I guess. I didn't realize that saying "event A has probability 1" was not the same as saying "A will definitely occur."

That is not a correct interpretation of the probability density function f(x) = 1/10. The value of a pdf at a particular x is not the probability that the value x will actually be realized.

By analogy, if the mass density of a 10 meter rod is 1/10 kg per meter, this does not mean that the weight of the rod "at" x = 3 meters is 1/10 kg.

Many rules of probability can be remembered by thinking of the pdf f(x) as giving "the probability of x", however this occasionally will lead to wrong conclusions. It's analogous to thinking of dy and dx in calculus as "infinitesimal numbers" That's useful, but it isn't an infallible way of reasoning.

Yeah, I wasn't thinking that through very well when I posted it. I guess what I meant was that for any interval of time with length n (for example, any given period of 1 minute) the probability was n/10 that the bus would show. I was confused because this doesn't seem to suggest the bus has to come at all. But you answered my question with your statement above. Thanks!