question about mean in probability?

Question

in probabilty distributions,how do we say the mean is given by the sum of the
product of the event and its probability.that is sum of xi.p(xi)value of i ranging from 1 to n.i want the physical interpretation

Answer 1

I assume that you are familiar with how to derive this result, so I won't go into that. Your question is interesting, I think that some of the confusion or uncertainty you may have may be a result of the muddled term "average" in this case. I have heard it called, and this would be my preferred way to name it since it seems more direct, the "expectation value".

It is not an average in the sense of say, average height (where you take a bunch of height measurements and average them to get to a sort of middle ground about which the others are scattered). In probability you're dealing with the likelihood of one event vs. another. In this sense when you "average" or take the expectation value what you are finding is the event that will most often come up (the one you expect, hence expectation value).

This should make sense from the definition: if you have an event with a high probability, it will figure largely in the calculation, vice versa for an event of low probability (in the same way that if you average heights for a group of people and you have 10 men that are 6 ft. tall and 1 that is 5 ft. tall, the average will be close to 6 ft. since there are so many at that height). So the expectation value will tend to cluster around things that have a high chance of occurring (and you do indeed see that events with high probability happen more often).

I hope that this helps, and I'm sorry that I couldn't give a more concise "physical" interpretation. Perhaps someone else can.

Answer 2

If each event had equal probabilities, you would simply add the events and divide by the number N of events. In effect you are giving each event an equal weighting of 1/N.

If each event has a different probability, you want to give each event a weighting based on its probability. That is why you multiply each event by its probability.