Ferkans — Interactive Telecom Tutor

Why Order Statistics?

Given $n$ i.i.d. random variables, the order statistics are the sorted values. The minimum and maximum are the simplest — and the most useful. In wireless communications, the SNR after selection combining is the maximum of $L$ i.i.d. channel gains, and the outage capacity is governed by the minimum eigenvalue of a random matrix. Understanding the distribution of extremes is essential.

Definition:
Order Statistics

Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with common CDF $F_{X}$ and PDF $f_{X}$ . The order statistics are the sorted values

$X_{(1)} \le X_{(2)} \le \cdots \le X_{(n)},$

where $X_{(1)} = \min(X_1, \ldots, X_n)$ and $X_{(n)} = \max(X_1, \ldots, X_n)$ .

Theorem: CDF of the Minimum and Maximum

For $n$ i.i.d. RVs with common CDF $F_{X}$ :

Maximum: $F_{X_{(n)}}(x) = [F_{X}(x)]^n$ .
Minimum: $F_{X_{(1)}}(x) = 1 - [1 - F_{X}(x)]^n$ .

Proof

Maximum

$F_{X_{(n)}}(x) = \mathbb{P}(\max_i X_i \le x) = \mathbb{P}(X_1 \le x, \ldots, X_n \le x) = \prod_{i=1}^n F_{X}(x) = [F_{X}(x)]^n,$ $

using independence.

Minimum

$\mathbb{P}(X_{(1)} > x) = \mathbb{P}(X_1 > x, \ldots, X_n > x) = [1 - F_{X}(x)]^n.$ $Therefore$ F_{X_{(1)}}(x) = 1 - [1 - F_{X}(x)]^n $.$ \blacksquare$

Theorem: PDF of the $k$ -th Order Statistic

The PDF of the $k$ -th order statistic $X_{(k)}$ from $n$ i.i.d. continuous RVs with CDF $F_{X}$ and PDF $f_{X}$ is

$f_{X_{(k)}}(x) = \frac{n!}{(k-1)!(n-k)!}\, [F_{X}(x)]^{k-1}\,[1 - F_{X}(x)]^{n-k}\,f_{X}(x).$

Proof

Counting argument

For $X_{(k)} \in (x, x+dx)$ , exactly $k-1$ of the $n$ RVs must fall below $x$ , exactly one must fall in $(x, x+dx)$ , and the remaining $n-k$ must exceed $x+dx$ . The multinomial coefficient counts the number of ways to assign these roles:

$f_{X_{(k)}}(x)\,dx \approx \frac{n!}{(k-1)!\,1!\,(n-k)!}\, [F_{X}(x)]^{k-1}\,[f_{X}(x)\,dx]\,[1-F_{X}(x)]^{n-k}.$

Dividing by $dx$ gives the result. $\blacksquare$

Example: Maximum of $n$ Uniform Random Variables

Let $X_1, \ldots, X_n$ be i.i.d. $\text{Uniform}[0,1]$ . Find the CDF and PDF of $M = \max(X_1, \ldots, X_n)$ and compute $\mathbb{E}[M]$ .

Solution

CDF

$F_{M}(x) = x^n$ for $x \in [0,1]$ .

PDF

$f_{M}(x) = n x^{n-1}$ for $x \in [0,1]$ .

Expected value

$\mathbb{E}[M] = \int_0^1 x \cdot n x^{n-1}\,dx = n \cdot \frac{1}{n+1} = \frac{n}{n+1}.$ $As$ n \to \infty $,$ \mathbb{E}[M] \to 1$ — the maximum concentrates near the upper endpoint.

Order Statistics of i.i.d. Uniforms

Visualize the PDF of the $k$ -th order statistic from $n$ i.i.d. $\text{Uniform}[0,1]$ random variables. Adjust $n$ and $k$ to see how the distribution shifts.

Parameters

n

(sample size)5

k

(order)5

Why This Matters: Selection Combining and Order Statistics

In a diversity receiver with $L$ independent branches, selection combining chooses the branch with the highest SNR. If the branch SNRs are i.i.d. $\text{Exp}(\bar{\gamma})$ (Rayleigh fading), then the post-combining SNR is $\gamma_{\text{SC}} = X_{(L)}$ , the maximum order statistic. Its CDF is $F_{\gamma_{\text{SC}}}(x) = (1 - e^{-x/\bar{\gamma}})^L$ . This directly gives the outage probability as a function of the number of diversity branches — a fundamental tradeoff in receiver design.

Common Mistake: Order Statistic Index Out of Range

Mistake:

Setting $k > n$ or $k < 1$ in the $k$ -th order statistic formula.

Correction:

The order statistic $X_{(k)}$ is only defined for $1 \le k \le n$ . The interactive plot should clamp $k$ to this range. Attempting $k > n$ produces a meaningless result.

Quick Check

If $X_1, \ldots, X_n$ are i.i.d. $\text{Exp}(\lambda)$ , what is the distribution of $X_{(1)} = \min(X_1, \ldots, X_n)$ ?

$\text{Exp}(n\lambda)$

$\text{Exp}(\lambda/n)$

$\text{Gamma}(n, \lambda)$

$\text{Exp}(\lambda)$

Correction:

\text{Exp}(n\lambda)

$\mathbb{P}(X_{(1)} > x) = (e^{-\lambda x})^n = e^{-n\lambda x}$ , so $X_{(1)} \sim \text{Exp}(n\lambda)$ .

Key Takeaway

The CDF of the maximum of $n$ i.i.d. RVs is $[F_{X}(x)]^n$ ; for the minimum it is $1 - [1 - F_{X}(x)]^n$ . The minimum of $n$ i.i.d. exponentials with rate $\lambda$ is again exponential with rate $n\lambda$ — a result used constantly in diversity analysis and reliability engineering.

Order Statistics