Order statistic

Let ${𝑋 𝑖} 𝑛 𝑖 = 1$ be real random variables. The $𝑗$ th order statistic $𝑋 (𝑗)$ is the $𝑗$ th smallest $𝑋 𝑖$ . #m/def/prob

Joint PDF

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

𝑓 ⃗ 𝐊 (⃗ 𝐤) = 𝑛! 𝑛 \prod 𝑗 = 1 𝑓 (𝑥 𝑗)

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

ℙ (𝑋 (𝑗) \leq 𝑥) = 𝑛 \sum 𝑘 = 𝑗 (𝑛 𝑘) 𝐹 (𝑥) 𝑘 (1 - 𝐹 (𝑥)) 𝑛 - 𝑘

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

𝑓 𝑋 (𝑗) = 𝑛 (𝑛 - 1 𝑗 - 1) 𝑓 (𝑥) 𝐹 (𝑥) 𝑗 - 1 (1 - 𝐹 (𝑥)) 𝑛 - 1

#state/develop | #lang/en | #SemBr