Probability: Difference between revisions
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===Joint Random Variables=== | ===Joint Random Variables=== | ||
Two random variables are independant | Two random variables are independant iff <math>f_{X,Y}(x,y) = f_X(x) f_Y(y)</math>.<br> | ||
Otherwise, the marginal distribution is <math>f_X(x) = \int f_{X,Y}(x,y) dy</math>. | Otherwise, the marginal distribution is <math>f_X(x) = \int f_{X,Y}(x,y) dy</math>. | ||
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Then the order statistics are <math>X_{(1)}, ..., X_{(n)}</math> where <math>X_{(i)}</math> represents the i'th smallest number. | 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: | 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_{(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> | <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: | If <math>X_i</math> has pdf <math>f</math>, the joint pdf of <math>X_{(1)}, ..., X_{(n)}</math> is: | ||
<math> | <math> | ||
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since there are n! ways perform a change of variables. | since there are n! ways perform a change of variables. | ||
===Individual PDF=== | |||
<math> | |||
f_{X(i)}(x) = \frac{n!}{(i-1)!(n-i)!} F(x)^{i-1} f(x) [1-F(x)]^{n-1} | |||
</math> | |||
==Inequalities and Limit Theorems== | ==Inequalities and Limit Theorems== | ||