Random Variables and Distributions - Applications

Random variable transformations and distribution relationships provide powerful tools for deriving new distributions from known ones. These connections simplify calculations and reveal deep mathematical structure.

Probability Integral Transform

Theorem

Let $X$ be a continuous random variable with CDF $F_X$ (assumed strictly increasing and continuous). Then: $U = F_X(X) \sim \text{Uniform}(0,1)$

Conversely, if $U \sim \text{Uniform}(0,1)$ , then: $X = F_X^{-1}(U)$ has CDF $F_X$ .

Proof: Let $U = F_X(X)$ . For $0 \leq u \leq 1$ : $P(U \leq u) = P(F_X(X) \leq u) = P(X \leq F_X^{-1}(u)) = F_X(F_X^{-1}(u)) = u$

This is the CDF of Uniform $(0,1)$ . □

Example

Inverse Transform Sampling: To generate samples from any distribution $F_X$ using a standard uniform random number generator:

Generate $U \sim \text{Uniform}(0,1)$
Return $X = F_X^{-1}(U)$

For $\text{Exponential}(\lambda)$ : $F_X(x) = 1 - e^{-\lambda x}$ , so: $X = F_X^{-1}(U) = -\frac{1}{\lambda} \ln(1-U)$

Since $1-U \sim \text{Uniform}(0,1)$ when $U$ does, we can use $X = -\frac{1}{\lambda} \ln U$ .

Order Statistics

Definition

Given random variables $X_1, \ldots, X_n$ , the order statistics $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$ are the values arranged in increasing order.

$X_{(1)} = \min\{X_1, \ldots, X_n\}$ (minimum)
$X_{(n)} = \max\{X_1, \ldots, X_n\}$ (maximum)
$X_{(k)}$ is the $k$ -th smallest value

Theorem

If $X_1, \ldots, X_n$ are IID with PDF $f$ and CDF $F$ , then $X_{(k)}$ has PDF: $f_{(k)}(x) = \frac{n!}{(k-1)!(n-k)!} F(x)^{k-1} [1-F(x)]^{n-k} f(x)$

Example

For $n$ uniform random variables on $[0,1]$ :

$X_{(1)}$ (minimum) has PDF $f_{(1)}(x) = n(1-x)^{n-1}$
$X_{(n)}$ (maximum) has PDF $f_{(n)}(x) = nx^{n-1}$
The median $X_{(\lceil n/2 \rceil)}$ has a more complex distribution

Box-Muller Transform

Theorem

If $U_1, U_2 \sim \text{Uniform}(0,1)$ are independent, then: $Z_1 = \sqrt{-2\ln U_1} \cos(2\pi U_2)$ $Z_2 = \sqrt{-2\ln U_1} \sin(2\pi U_2)$

are independent standard normal random variables: $Z_1, Z_2 \sim \mathcal{N}(0,1)$ .

This elegant result provides a method to generate normal random variables from uniform ones.

Reliability and Hazard Functions

Definition

For a non-negative random variable $X$ (e.g., lifetime):

The reliability function (survival function) is: $R(t) = P(X > t) = 1 - F_X(t)$

The hazard function (failure rate) is: $h(t) = \frac{f_X(t)}{R(t)} = \frac{f_X(t)}{1 - F_X(t)}$

The hazard function represents the instantaneous failure rate given survival to time $t$ .

Example

For $X \sim \text{Exponential}(\lambda)$ : $R(t) = e^{-\lambda t}, \quad h(t) = \frac{\lambda e^{-\lambda t}}{e^{-\lambda t}} = \lambda$

The constant hazard rate reflects the memoryless property: the failure rate doesn't depend on age.

For the Weibull distribution with shape parameter $\beta$ :

$\beta < 1$ : decreasing failure rate (infant mortality)
$\beta = 1$ : constant failure rate (exponential distribution)
$\beta > 1$ : increasing failure rate (wear-out failures)

Remark

These transformations and relationships form a toolkit for both theoretical analysis and practical simulation. The probability integral transform enables Monte Carlo simulation, order statistics are central to non-parametric statistics, and hazard functions are fundamental in reliability engineering and survival analysis.