Multivariate random variable

Order statistic

Let {𝑋𝑖}𝑛𝑖=1 be real random variables. The 𝑗th order statistic 𝑋(𝑗) is the 𝑗th smallest 𝑋𝑖. prob

Joint PDF

If {𝑋𝑖}𝑛𝑖=1 are identically and independently distributed with probability density 𝑓, then the Joint probability density function of ⃗𝐊 =(𝑋(𝑗))𝑛𝑗=1 is

𝑓⃗𝐊(⃗𝐤)=𝑛!𝑛∏𝑗=1𝑓(𝑥𝑗)

Marginal CDF

If {𝑋𝑖}𝑛𝑖=1 are identically and independently distributed with Cumulative distribution function 𝐹, then the marginal CDF of 𝑋(𝑗) is

ℙ(𝑋(𝑗)≤𝑥)=𝑛∑𝑘=𝑗(𝑛𝑘)𝐹(𝑥)𝑘(1−𝐹(𝑥))𝑛−𝑘

Marginal PDF

If {𝑋𝑖}𝑛𝑖=1 are identically and independently distributed with Cumulative distribution function 𝐹 and probability density 𝑓 then the marginal probability density of 𝑋(𝑗) is

𝑓𝑋(𝑗)=𝑛(𝑛−1𝑗−1)𝑓(𝑥)𝐹(𝑥)𝑗−1(1−𝐹(𝑥))𝑛−1

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