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

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In summary: Just as there is a central limit theorem for the sample mean, there is a similar result for order statistics.
  • #1
cloud360
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Probability.What does: "max{X1,X2, . . . ,Xn}", mean

I don't know what the underlined part is saying, or what it is asking

Homework Statement



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.

Homework Equations



none

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?
 
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  • #2


When observations are taken [tex] Y [/tex] is the largest of those observations.
 
  • #3


cloud360 said:
Does this mean Y represents only 1 single discrete random variable from this set: {X1,X2, . . . ,Xn}.
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.
 
  • #4


D H said:
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.

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?
 
  • #5


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]
 
  • #6


micromass said:
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]


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))
 
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  • #7


cloud360 said:
this has really confused me a lot. please can you clarrify.
I think D H interpreted your question
cloud360 said:
Does this mean Y represents only 1 single discrete random variable from this set: {X1,X2, . . . ,Xn}.
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.
 
  • #8


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.
 

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

What is probability?

Probability is a measure of the likelihood that a specific event will occur. It is typically expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

How is probability calculated?

Probability is calculated by dividing the number of desired outcomes by the total number of possible outcomes. This can be represented as a fraction, decimal, or percentage.

What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual observations and can vary depending on the sample size.

What is the meaning of "max{X1,X2,.,Xn}" in probability?

In probability, "max{X1,X2,.,Xn}" refers to the maximum value among a set of random variables. It represents the highest possible outcome among a given group of events.

How does the concept of probability apply in scientific experiments?

In scientific experiments, probability is used to predict the likelihood of a certain outcome or to make decisions based on the likelihood of different outcomes. It is also used to determine the significance of results and to draw conclusions from data.

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