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I have three independent random variables, $X_1$, $X_2$, and $X_3$, and each random variable has different mean. (And I assume they have the same variance). I would like to know how I can get the probability that $X_1$ is the max of all three. And help would be appreciated.

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  • $\begingroup$ Do you know the distributions? $\endgroup$ Commented May 16, 2016 at 18:23
  • $\begingroup$ I assume they all have the normal distributions with same variance but different means. $\endgroup$
    – Jae Chung
    Commented May 16, 2016 at 18:29
  • $\begingroup$ Do values of X1 correspond to values of X2 & X3? $\endgroup$ Commented May 16, 2016 at 18:30
  • $\begingroup$ gung, I don't understand your question. Could you clarify? $\endgroup$
    – Jae Chung
    Commented May 16, 2016 at 18:39
  • $\begingroup$ Closely related: stats.stackexchange.com/questions/87241 (generic formulation), stats.stackexchange.com/questions/173064 (two independent normals), stats.stackexchange.com/questions/139072 (two correlated normals), etc. Note that $X_2-X_1$ and $X_3-X_1$ are correlated Normals and $X_1$ is the max of all $X_i$ if and only if the larger of these two differences is non-positive. Thus, you may directly apply the answer in the last link. $\endgroup$
    – whuber
    Commented May 16, 2016 at 20:04

1 Answer 1

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Assume their means are $\mu_1,\mu_2,\mu_3$, with the same standard deviation, $\sigma$. Then the probability that the first random variable is the max is:

$$P(X_1>X_2,X_1>X_3)=\int_{-\infty}^\infty P(x>X_2|X_1=x)P(x>X_3|X_1=x)f(x)dx$$

where $f(x)$ is the density of $X_1$. It's doubtful whether you can simply this further.

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  • $\begingroup$ Since the variables are independent this simplifies to $\int_{-\infty}^{\infty} {P(X_2 < x) P(X_3 < x) f(x) dx}$ $\endgroup$
    – jodag
    Commented Apr 24, 2023 at 1:43

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