Largest of 3 random variables

In summary, the problem is to find the probability that the largest of a set of n identically distributed random variables is greater than a given random variable. The problem can be solved by using order statistics.
  • #1
4
0
Hi,

I am stuck with the problem of solving this problem for my research.

I have 3 random variables say X, Y, and Z and say
Pr[X > Y] = p_xy, Pr[X > Z] = p_xz, and Pr[Y > Z] = 0.5.

Note that p_yx = 1 - p_xy. Similarly, p_zx = 1 - p_xz, p_yz = p_zy = 0.5

I need to find out the Pr[X > Y and X > Z] ?

Any directions to solve this problem would be of great help.
 
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  • #2
It depends on whether or not Y and Z are independent. If they are:
P(X>Y and X>Z) = P(X>Y)P(X>Z) = (p_xy)(p_xz).
If they are not independent then there is no easy answer.
 
  • #3
they are dependent variables.

Is there any way I can use ranking schemes that are used in Information Retrieval ?
 
  • #4
chetkgp said:
they are dependent variables.

Is there any way I can use ranking schemes that are used in Information Retrieval ?

You don't have any more refined information about the probability density functions?
 
  • #5
To be more specific :

I have N normal distributions, all having exactly the same mean
except for one (say X). The variance is same in all cases.

I know the probability Pr(X > Y_i), for any of the N-1 remaining distributions.
Using this I need to know the probability that X is the largest of all the distributions.

-Chester
 
  • #6
chetkgp said:
To be more specific :

I have N normal distributions, all having exactly the same mean
except for one (say X). The variance is same in all cases.

I know the probability Pr(X > Y_i), for any of the N-1 remaining distributions.
Using this I need to know the probability that X is the largest of all the distributions.

-Chester

Have you come across order statistics?
 
  • #7
I just read about order statistics.
I'm not sure how to apply it here because the distributions are not identical.

-Chester
 
  • #8
chetkgp said:
I just read about order statistics.
I'm not sure how to apply it here because the distributions are not identical.

-Chester

You can modify those formulas for distributions that are different, as long as they are independent, so maybe you won't be able to do it.

What you should do is look at the formulation of the order statistic distribution. Look at the first statements for the derivation and see if you can either a) use those assumptions or b) create a modified version of the order statistic distribution to suit your needs.

You can create order statistics that use different distributions, but the common formula was derived with the assumption that you use the same distribution.

I have a feeling the modification won't be simple if you do find a distribution for your ordered statistic given your distributions, but it might be worth investigating particularly if your problem is important.
 
  • #9
chetkgp,

You didn't say whether you need a 1) A symbolic answer or 2) A numerical answer that is easy to justify (e.g. one that comes from a symbolic expression and a well known deterministic approximation method for it) or 3) A practical numerical answer, which could come from a Monte-Carlo simulation.

You can think of the problem this way:

Let the independent random variables be [itex] X, X_1,X_2,...X_{n-1} [/itex] with [itex] X_1,X_2,..X_{n-1} [/itex] being identically distributed. Compute the distribution of [itex] Y = max \{X_1,X_2,..X_{n-1}\} [/itex] using the formulae from the theory of order statistics.

Then find the probabiity that [itex] X > Y [/itex].
 

1. What is the largest of 3 random variables?

The largest of 3 random variables refers to the maximum value among the three variables. It is the largest number that can be obtained from the three randomly chosen variables.

2. How is the largest of 3 random variables calculated?

The largest of 3 random variables is calculated by comparing the three variables and selecting the one with the highest value. This can be done manually or using programming techniques such as loops and conditional statements.

3. Can the largest of 3 random variables be negative?

Yes, the largest of 3 random variables can be negative if one or more of the three variables have negative values. The largest variable will still be the one with the highest absolute value, regardless of its sign.

4. How is the largest of 3 random variables useful in scientific research?

The largest of 3 random variables can be useful in analyzing and comparing data sets in scientific research. It can help identify outliers and extreme values, and can also be used in statistical analysis to determine the maximum value of a set of data.

5. Is there a difference between the largest of 3 random variables and the largest of 3 fixed variables?

Yes, there is a difference between the largest of 3 random variables and the largest of 3 fixed variables. Random variables are chosen randomly and can have different values each time, while fixed variables have predetermined values. Therefore, the largest value among 3 fixed variables will always be the same, while the largest of 3 random variables can vary each time they are chosen.

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