Hi, In the 2nd exam question for point estimation, we are asked to find the distn of XMax. The result looks at P(XMax<=x)=P(All Xi<=x). I'm missing why why we are looking at XMax<=x rather than XMax>=x. Maybe just tired but can someone please explain - thanks. Stewart
Often in problems of this type, it can be very difficult to work out in one direction and much easier in the other. How would you write down the equivalent case for P(Xmax>x)
Well I am assuming (wrongly it would seem) that P(Xmax>x) is the probability that the maximum values of X are greater than all the other values of X but I'm sure I'm getting my X's and x's mixed up somewhere.
Put it this way: the set of events that make up Xmax<x is identical to the set of events that all the xi<x. They are the same thing. However, Xmax>x doesn't mean all xi>x; it means at least one xi>x. Then you have to work out all the combinations of xi>x and xj<x for all i,j in X and assign a probability to each. It's doable but tedious and error prone. It's very common in the kinds of problem that it's easier to work out the complement of an event than the event itself.
Calum's right. Think of a list of 5 numbers: 3, 6, 9, 12, 20 The maximum is 20. If we say the maximum is less than 30 - then we know that all the other numbers are also less than 30. Whereas if we say the maximum is greater than 10 - then we are not sure whether any of the other numbers also are involved. Similarly if you are working with a minimum then using greater than probabilities will make life easier. A great exam question on this was Subject 101, April 2002, Q9.
My question may even be at a simpler level than that. I am struggling to get my head around the distn of XMax. I am assuming this is saying that across all samples we take, if we took the maximum value, these maximum values would form the distn of XMax. The exam solution says "the value of XMax will be less than some value x.." - what is "x" and why is XMax less than x when XMax is a maximum - shouldn't it be bigger than everything? Sorry Guys, I'm sure this is a fairly basic question but I'm missing some basic understanding here and I'm keen to get each of the foundations right. Thanks for your help. Stewart.
When we're chatting about a distribution then we're looking at the probabilities of all the values it can take. Hence, we'll look at P(XMax <= x) for any x. The confusion you're having is you're thinking of a particular sample rather than a theoretical distribution. hence you say "it is the maximum of the sample values" rather than "what values could it take in the future". Does that help?
That makes sense what your saying about a theoretical distribution. I'm just not getting we we are saying <= rather than >. What Calum says about taking the compliment also makes sense but then wouldn't it be 1-P(XMax<x) but the result doesn't do that. It says distn of XMax is [F(x)]^n.
I for one find that thinking about the distribution of random variables that are derived from other random variables can get mindbending very, very quickly. One thing to do is keep in mind the fundamental principles of random variables in general. Xmax, whatever it is, is a random variable, like any other (it just happens to depend on other random variables). It has a range of values it can take and you can assign a probability [density] to those values. To construct the probability of any particular value of Xmax, you need to consider the probabilities of the possible ways to produce that value. In this case, for Xmax to be equal or less than some particular number (say 3), all the random variables of which Xmax is the max have to be less than three. If they are independent, then the probability is simply the product of the probability of the individual variables, which is where your F(x)^n comes from.
I think a few steps closer on this one so I'll leave a bookmark to come back to it as I get my head better around it. Thanks for your help.
