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

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

    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?????
     
    Last edited: Nov 1, 2010
  2. jcsd
  3. statdad

    statdad 1,467
    Homework Helper

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

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

    Staff: Mentor

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

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

    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????
     
  6. micromass

    micromass 18,699
    Staff Emeritus
    Science Advisor
    Education Advisor

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

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


    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))
     
    Last edited: Nov 1, 2010
  8. vela

    vela 12,561
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

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

    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 Xi that happens to be the maximum, which implies that Y obeys the same statistics as the X's do.
     
  9. D H

    Staff: Mentor

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

    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.
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?