Probability.What does: "max{X1,X2, . . . ,Xn}", mean

cloud360

I dont know what the underlined part is saying, or what it is asking
1. The problem statement, all variables and given/known data

For n >= 1, let X1,X2, . . . ,Xn denote n independent and identically distributed random variables
according to X, with cdf FX(y). Let Y = max{X1,X2, . . . ,Xn}.

Show that the cdf FY (y) of Y
satisfies
FY (y) = (FX(y))^n .................y in the set R.

2. Relevant equations

none

3. The attempt at a solution
I think Y = max{X1,X2, . . . ,Xn} means the maximum discrete random variable in that set. And Y takes the maximum value.

Does this mean Y represents only 1 single discrete random variable from this set: {X1,X2, . . . ,Xn}.


Am i correct?????

statdad

When observations are taken [tex] Y [/tex] is the largest of those observations.

D H

Not really.

Suppose you have an experiment that involves taking 100 different measurements of what is presumably the same thing, with each measurement presumable independent of the other measurements. In short, you have a set of 100 independent and identically distributed (i.i.d.) random variables.

You run the experiment and find (for example) that the 56th value happens to be the largest. You run the experiment again, this time finding that the 11th value happens to be the largest. The next time through it is the 89th that is the largest, and so on. This largest value is itself a random variable. The statistics that describe this random variable are different from but related to the statistics than describes the underlying random process.

You can also investigate the behavior of the second largest value, or the third largest, etc. This is the world of order statistics, a very powerful statistical technique.

cloud360

this has really confused me a lot. please can you clarrify.

when you say "not really", are you reffering to the case where each random variable: x1,x2,x3 is equal to each other. i.e they have same value.

For example if we look at 100 measurements and we find that each 1 is indetical/same e.g each one = 73cm, e.t.c??

This means y can have more than 1 value? e.g if 10 of the variables are identical/73cm....and 73cm is the highest size. This means y takes 10 values????

micromass

Let every [tex]X_i:\Omega\rightarrow \mathbb{R} [/tex]. Then [tex]Y=\max{\{X_1,...,X_n\}}[/tex] is defined as

[tex]Y:\Omega\rightarrow \mathbb{R}:\omega\rightarrow \max\{X_1(\omega),...,X_n(\omega)\}[/tex]

cloud360

isnt the "w the same thing as omega????

are you saying that Max(Fx(2.1)), is asking for the cdf of Px(1)+Px(2).


Like how Fx(3.2)=Px(1)+Px(2)+Px(3).....are you saying this is same as max(Fx(3.2))

Like how Fx(1.2)+Fx(3.2)=Px(1)+Px(2)+Px(3).....................are you saying this is same as max(Fx(3.2)+Fx(1.2))=Max(Fx(3.2))

vela

I think D H interpreted your question
differently from how you meant it because your question is pretty unclear. What exactly do you mean by "represents" when you ask if "Y represents only 1 single discrete random variable"? It sounds like you're suggesting Y is a label given to the particular X_i that happens to be the maximum, which implies that Y obeys the same statistics as the X's do.

D H

That is exactly how I interpreted his question. The distribution for some order statistic is related to but different from the distribution that describes the Xs.

Note that the same is true for the sample mean, [itex]\frac 1 n \sum_i x_i[/itex]. It too is a random variable with its own statistics.