2 Random variables

A random variable $X$ is a quantity whose value cannot be predicted with certainty. We assume that for every real number $a$ the probability $\text{P}\left(X=a\right)$ in a trial is well-defined. In practice, engineers are often concerned with two broad types of variables and their probability distributions: discrete random variables and their distributions, and continuous random variables and their distributions. Discrete distributions arise from experiments involving counting, for example, road deaths, car production and aircraft sales, while continuous distributions arise from experiments involving measurement, for example, voltage, corrosion and oil pressure.

2.1 Discrete random variables and probability distributions

A random variable $X$ and its distribution are said to be discrete if the values of $X$ can be presented as an ordered list say $x_1,x_2,x_3,\dots$ with probability values $p_1,p_2,p_3,\dots$ . That is $\text{P}\left(X=x_i\right)=p_i$ . For example, the number of times a particular machine fails during the course of one calendar year is a discrete random variable.

More generally a discrete distribution $f\left(x\right)$ may be defined by:

$f\left(x\right)=\left\{\begin{array}{cc}\hfill p_i\hfill & \hfill \text{if}x=x_{i}i=1,2,3,\dots \hfill \\ \hfill 0\hfill & \hfill \text{otherwise}\hfill \end{array}\right.$

The distribution function $F\left(x\right)$ (sometimes called the cumulative distribution function) is obtained by taking sums as defined by

$F\left(x\right)={\sum }_{x_i\le x}f\left(x_i\right)={\sum }_{x_i\le x}{p}_i$

We sum the probabilities $p_i$ for which $x_i$ is less than or equal to $x$ . This gives a step function with jumps of size $p_i$ at each value $x_i$ of $X$ . The step function is defined for all values, not just the values $x_i$ of $X$ .

Example 5

Turbo Generators plc manufacture seven large turbines for a customer. Three of these turbines do not meet the customer’s specification. Quality control inspectors choose two turbines at random. Let the discrete random variable $X$ be defined to be the number of turbines inspected which meet the customer’s specification.

1. Find the probabilities that $X$ takes the values $0,1$ or $2$ .
2. Find and graph the cumulative distribution function.

Solution

1. The possible values of $X$ are clearly $0,1$ or $2$ and may occur as follows:

   Sample Space	Value of X
   Turbine faulty, Turbine faulty	0
   Turbine faulty, Turbine good	1
   Turbine good, Turbine faulty	1
   Turbine good, Turbine good	2

   We can easily calculate the probability that $X$ takes the values $0,1$ or $2$ as follows:

   $$\begin{array}{llll}\hfill \text{P}\left(X=0\right)& =\frac{3}{7}\times \frac{2}{6}=\frac{1}{7}& \hfill & \\ \hfill \text{P}\left(X=1\right)& =\frac{4}{7}\times \frac{3}{6}+\frac{3}{7}\times \frac{4}{6}=\frac{4}{7}& \hfill & \\ \hfill \text{P}\left(X=2\right)& =\frac{4}{7}\times \frac{3}{6}=\frac{2}{7}& \hfill & \end{array}$$

   The values of $F\left(x\right)={\sum }_{x_i\le x}\text{P}\left(X=x_i\right)$ are clearly

   $F\left(0\right)=\frac{1}{7}F\left(1\right)=\frac{5}{7}\text{and}F\left(2\right)=\frac{7}{7}=1$

2. The graph of the step function $F\left(x\right)$ is shown below.

   Figure 1