Continuous Probability Distributions

Chickentown

Hi,

I am trying to get my head around continuous probability distributions and I thought if I posted some of my presumptions up here some of you might critique me and this may help me to understand the topic better.

So this whole business of continuous probability distributions; I am attempting to understand the concept by visualising the arrival time for a train (T_AVG) on some hypothetical platform A which is due, lets say, at 2pm each day.

The train may arrive early at 1:59pm, late at 2:01pm or even later at 2:15pm, in theory the train my never turn up and therefore be ''lateness'' goes off to infinity.
And each day this has a chance of happening. So if we knew T_AVG, ad we know that like most things train arrival times are normally distributed.

We might want to know things like: What is the the greater probability?; Train arriving at 2:01 or 2:15, or between 2:05-2:10 or before 2:30 or whatever..

Or if we were going to take a journey that has two possible routes (each route take exactly the same time to travel) and we don't mind paying the higher fare for route B so long as we know that there is a higher probability that route A will be later (delayed longer). We could compare both distributions using the standard normal distribution.

This is pretty much where I am at so far with the overall picture of what continuous distributions can help us to do.

Any thoughts, criticism, tests to my understanding welcome.

Thank you

dlouth15

I think that is a correct understanding of probability distributions.

You could use them to compare train routes if you had such information. If not being late for a meeting was of paramount importance, you would choose the route with the less spread-out distribution - less chance of the train overrunning. You might even choose a train with a later arrival time if its probability distribution was sufficiently narrow.

The only thing I would say is that the probability distribution won't tell you the probability of arriving at precicely 2.01. That will always be zero.

Instead it will tell you the probability of the train arriving within a range of times. So 2.01±2 minutes would be a valid range as would 2.00-2.10 and so on. Then the probability would be the area under the curve within that range.

Other valid ranges would be, for example, "later than 2.01" or "on or before 2.00", etc. Again the probability of these would be done by calculating the area under the curve. In first case it would be the area under the curve from 2.01 to +infinity.

Michael Collins

dlouth15 wrote: »

...The only thing I would say is that the probability distribution won't tell you the probability of arriving at precicely 2.01. That will always be zero...

Exactly - I think this is an important point. It might seem a bit strange that the probability of it arrving a certain time is zero, but remember, if you're asking does it arrive at 2:01pm, you're actually asking, does it arrive at exactly 2 hr 0 m 0 sec 0 milliseconds all the way down to 0 nanoseconds and so on... So the chances of it arriving at such a precise instant in time, exactly, will be very small, and effectively zero.

That's why, as dlouth15 says, it makes sense instead to talk of ranges of something - time in this case.

Note that discrete probability distributions don't have this issue.