Inequality Truth: X1 < X2 & X > 0

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In summary, the conversation discusses the comparison between two inequalities involving random variables and the sum of their largest values. It is established that the inequality is always true, but there is a possibility for the probabilities to be equal in some cases. This is illustrated with an example of a random variable that always takes the value of 1. The conversation also mentions a fact about the sum of the largest values being greater than or equal to the arithmetic mean multiplied by the number of values.
  • #1
EngWiPy
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Hi,

Is the following inequality true for x>0:

Pr[X1<x]<Pr[X2<x] for X1<X2?
 
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  • #2
No, they can be equal for some distributions.
 
  • #3
disregardthat said:
No, they can be equal for some distributions.

Ok, suppose we have n independent and identically distributed random variables x1,x2,...,xn. There is a fact that for positive numbers, the sum of the largest L<n numbers is greater than or equal the arithmetic mean multiplied by L, i.e.:

[tex]\sum_{i=1}^Lx_{(i)}\geq\frac{L}{n}\sum_{i=1}^nx_i[/tex]

where [tex]x^{(i)}[/tex] are the order statistics in descending order. Then is it true to say that:
[tex]\text{Pr}\left[\sum_{i=1}^Lx_{(i)}<x\right]\leq\text{Pr}\left[\frac{L}{n}\sum_{i=1}^nx_i<x\right][/tex]
 
  • #4
Yes, that's always true. What disregardthat was pointing out is simply that the probabilities might be equal, which you have in this post but didn't have in your OP. For example consider the stupid random variable which always takes the value of 1. Then the sum of the L largest is L, and L/n*mean is L as well. so your probabilities are either both 0 or both 1 depending on what the value of x is
 
  • #5
Office_Shredder said:
Yes, that's always true. What disregardthat was pointing out is simply that the probabilities might be equal, which you have in this post but didn't have in your OP. For example consider the stupid random variable which always takes the value of 1. Then the sum of the L largest is L, and L/n*mean is L as well. so your probabilities are either both 0 or both 1 depending on what the value of x is

I forgot to include it. Thanks
 

1. What is the meaning of "Inequality Truth: X1 < X2 & X > 0"?

The statement "Inequality Truth: X1 < X2 & X > 0" means that X1 is less than X2 and X is greater than 0. This is known as a compound inequality, where two inequalities are combined using the "and" symbol (&). In this case, it is stating that X1 is a smaller value than X2 and X is a positive number.

2. Why is "Inequality Truth: X1 < X2 & X > 0" important in scientific research?

Compound inequalities like "Inequality Truth: X1 < X2 & X > 0" are important in scientific research because they allow us to compare and analyze different variables. This statement can help us understand the relationship between X1 and X2, as well as the range of values that X can take on. It can also be used to set boundaries or limitations in experiments and studies.

3. How is "Inequality Truth: X1 < X2 & X > 0" used in data analysis?

In data analysis, "Inequality Truth: X1 < X2 & X > 0" can be used to filter and sort data. For example, if we have a data set with two variables, X1 and X2, we can use this statement to only analyze data points where X1 is smaller than X2 and X is greater than 0. This can help us identify patterns and trends in the data.

4. What does it mean when X1 is less than X2?

X1 being less than X2 means that X1 has a smaller numerical value than X2. This is represented by the < symbol in the compound inequality. For example, if X1 = 3 and X2 = 5, then X1 < X2 is true because 3 is less than 5.

5. How can we apply "Inequality Truth: X1 < X2 & X > 0" in real-world scenarios?

Compound inequalities like "Inequality Truth: X1 < X2 & X > 0" can be applied in many real-world scenarios, such as in economics, finance, and social sciences. For example, it can be used to compare the incomes of different groups or to analyze the relationship between two variables in a study. It can also be used to set conditions or requirements in various fields, such as determining eligibility for a loan or scholarship.

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