# Largest of 3 random variables

## Main Question or Discussion Point

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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mathman
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.

they are dependent variables.

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

chiro
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?

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

chiro
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?

I'm not sure how to apply it here because the distributions are not identical.

-Chester

chiro
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.

Stephen Tashi
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 $X, X_1,X_2,...X_{n-1}$ with $X_1,X_2,..X_{n-1}$ being identically distributed. Compute the distribution of $Y = max \{X_1,X_2,..X_{n-1}\}$ using the formulae from the theory of order statistics.

Then find the probabiity that $X > Y$.