Concept of a Random Variable

• It is often important to allocate a numerical description to the outcome of a statistical experiment.
• These values are random quantities determined by the outcome of the experiment.
• Definition 3.1:
  
• We use a capital letter, say $X$, to denote a random variable and its corresponding small letter , $x$ in this case, for one of its value.
• One and only one numerical value is assigned to each sample point $X$.

• Example 3.1: Two balls are drawn in succession without replacement from an box containing 4 red balls and 3 black balls.
• The possible outcomes and the values y of the random variable Y, where Y is the number of red balls, are

Sample
Space	y
RR	2
RB	1
BR	1
BB	0

Example: Number of defective (D) products when 3 products are tested.

Outcomes in	$x$: value
Sample Space	of $X$
DDD	3
DDN	2
DND	2
DNN	1
NDD	2
NDN	1
NND	1
NNN	0

• Example 3.3: Components from the production line are defective or not defective.
• Define the random variable $X$ by
  $$X=\left\lbrace \begin{array}{cr} 1, & if  the  component  ... ...e  component  is  not  defective \\ \end{array}\right\rbrace$$

• This random variable is categorical in nature.

• Example 3.5: A process will be evaluated by sampling items until a defective item is observed.
• Define $X$ by the number of consecutive items observed

Sample
Space	x
D	1
ND	2
NND	3
&vellip#vdots;	&vellip#vdots;

• According to the countability of the sample space which is measurable, it can be either discrete or continuous.
• Discrete random variable: If a random variable take on only a countable number of distinct values.
  • If the set of possible outcomes is countable
  • Often represent count data, such as the number of defectives, highway fatalities
• Continuous random variable: If a random variable can take on values on a continuous scale.
  • often represent measured data, such as heights, weights, temperatures, distance or life periods
• Definition 3.2:
  
• Definition 3.3: