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→Order Statistics
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==Order Statistics== | ==Order Statistics== | ||
Consider iid random variables <math>X_1, ..., X_n</math>.<br> | |||
Then the order statistics are <math>X_{(1)}, ..., X_{(n)}</math> where <math>X_{(i)}</math> represents the i'th smallest number. | |||
;Min and Max | |||
The easiest to reason about are the minimum and maximum order statistics: | |||
<math>P(X_{(1)} <= x) = P(\text{min}(X_i) <= x) = 1 - P(X_1 > x, ..., X_n > x)</math> | |||
<math>P(X_{(n)} <= x) = P(\text{max}(X_i) <= x) = P(X_1 <= x, ..., X_n <= x)</math> | |||
;Joint PDF | |||
If <math>X_i</math> has pdf <math>f</math>, the joint pdf of <math>X_{(1)}, ..., X_{(n)}</math> is: | |||
<math> | |||
g(x_1, ...) = n!*f(x_1)*...*f(x_n) | |||
</math> | |||
since there are n! ways perform a change of variables. | |||
;Individual PDF | |||
==Inequalities and Limit Theorems== | ==Inequalities and Limit Theorems== | ||