STAT 3360 Notes

1. Events
2. References

1 Events

1.1 Random Experiment

A random experiment is an action or process that leads to one of several possible outcomes.
The outcomes of an experiment are represented by values of the relevant variable.
The experiment can be about a single variable or about two (or more) variables.

1.2 Random Experiment: One Variable

If the experiment is about a single varibale, then the outcomes are possible values of the variable.

1.2.1 Examples

if the experiment is about a single variable Height, then the outcomes can be "tall", "medium" or "short".

Varibale outcome 1 outcome 2 outcome 3

Height tall medium short
if the experiment is about a single variable Weight, then the outcomes can be "heavy", "medium" or "light".

Varibale outcome 1 outcome 2 outcome 3

Weight heavy medium light

Varibale	outcome 1	outcome 2	outcome 3
Height	tall	medium	short

Varibale	outcome 1	outcome 2	outcome 3
Weight	heavy	medium	light

1.3 Random Experiment: Two Variables

If the experiment is about two (or more) variables, then the outcomes are the combinations of the values of the two (or more) variables.

1.3.1 Example

For example, if the experiment is about two variables Height and Weight (ie, about (Height, Weight)), then

"height = tall" is NOT an outcome of this experiment because the experiment is about both Height and Weight while "height = tall" doesn't give information about Weight.

the outcome of this experiment should be (tall, heavy) or (tall, medium weight) or (tall, light) or (medium height, heavy) or (medium height, medium weight) or (medium height, light) or (short, heavy) or (short, medium weight) or (short, light).

Outcome	heavy	medium	light
tall	(tall, heavy)	(tall, medium weight)	(tall, light)
medium	(medium height, heavy)	(medium height, medium weight)	(medium height, light)
short	(short, heavy)	(short, medium weight)	(short, light)

1.4 Simple Event

An individual outcome of an experiment is called a simple event.

1.4.1 Example: A Single Variable

If the experiment is about a single variable Height, then
- "the student is tall" is a simple event
  
  Height tall medium short
  
  The Event \(\checkmark\)
- "the student is medium in height" is a simple event
  
  Height tall medium short
  
  The Event \(\checkmark\)
- "the student is short" is a simple event
  
  Height tall medium short
  
  The Event \(\checkmark\)

Height	tall	medium	short
The Event	\(\checkmark\)

Height	tall	medium	short
The Event		\(\checkmark\)

Height	tall	medium	short
The Event			\(\checkmark\)

1.4.2 Example: Two Variables

If the experiment is about both Height and Weight, then
- "the student is tall and heavy" is a simple event of the experiment
  
  The Event heavy medium light
  
  tall \(\checkmark\)
  
  medium
  
  short
- "the student is tall and meidum in weight" is a simple event of the experiment
  
  The Event heavy medium light
  
  tall \(\checkmark\)
  
  medium
  
  short
- …
- there are totally 9 simple events in this example.

1.5 Event

An event is a set of one or more simple events of an experiment.

1.5.1 Example: A Single Variable

If the experiment is about a single variable Height, then
- "the student is tall" is a simple event and thus an event
  
  Height tall medium short
  
  The Event \(\checkmark\)
- "the student is tall or short" is an event but not a simple event because it consists of more than 1 simple events
  
  Height tall medium short
  
  The Event \(\checkmark\) \(\checkmark\)

Height	tall	medium	short
The Event	\(\checkmark\)

Height	tall	medium	short
The Event	\(\checkmark\)		\(\checkmark\)

1.5.2 Example: Two Variables

If the experiment is about both Height and Weight, then
- "the student is tall and heavy" is a simple event and thus an event
  
  The Event heavy medium light
  
  tall \(\checkmark\)
  
  medium
  
  short
- "the student is tall and not heavy" is an event but not a simple event because it consists of more than 1 simple events
  
  The Event heavy medium light
  
  tall \(\checkmark\) \(\checkmark\)
  
  medium
  
  short

1.6 Complement Event

The complement of event A is the event that occurs when event A does not occur, which is denoted by A^C.
The complement event can be either a simple event or not, which depends on the situation.

1.6.1 Example: A Single Variable

If the experiment is about a single variable Height, then
- the complement of A = "the student is tall" is
  
  A^C = "the student is not tall" = "the student is short or medium"
  
  Height tall medium short
  
  The Event A A^C A^C
- the complement of A = "the student is tall or short" is
  
  A^C = "the student is not tall and not short" = "the student is medium in height"
  
  Height tall medium short
  
  The Event A A^C A

Height	tall	medium	short
The Event	A	A^C	A^C

Height	tall	medium	short
The Event	A	A^C	A

1.6.2 Example: Two Variables

If the experiment is about both Height and Weight, then
- the complement of A = "the student is tall and heavy" is
  
  A^C = "the student is not tall or not heavy"
  
  The Event heavy medium light
  
  tall A A^C A^C
  
  medium A^C A^C A^C
  
  short A^C A^C A^C
- the complement of A = "the student is tall or heavy" is
  
  A^C = "the student is not tall and not heavy"
  
  The Event heavy medium light
  
  tall A A A
  
  medium A A^C A^C
  
  short A A^C A^C

The Event	heavy	medium	light
tall	A	A^C	A^C
medium	A^C	A^C	A^C
short	A^C	A^C	A^C

The Event	heavy	medium	light
tall	A	A	A
medium	A	A^C	A^C
short	A	A^C	A^C

1.7 Intersection of Events

The intersection of events A and B is the event that occurs when both A and B occur, which is denoted by [ A and B ].
The intersection event can be either a simple event or not, which depends on the situation.

1.7.1 Example: A Single Variable

If the experiment is about a single variable Height, then
- the intersection between A = "the student is tall" and B = "the student is short" is
  
  [ A and B ] = "the student is both tall and short" = "impossible"
  
  Height tall medium short
  
  Event A \(\checkmark\)
  
  Event B \(\checkmark\)
  
  Event [ A and B ] \(\times\) \(\times\) \(\times\)
- the intersection between A = "the student is tall" and B = "the student is not short" is
  
  [ A and B ] = "the student is tall and not short" = "the student is tall"
  
  Height tall medium short
  
  Event A \(\checkmark\)
  
  Event B \(\checkmark\) \(\checkmark\)
  
  Event [ A and B ] \(\checkmark\) \(\times\) \(\times\)
- the intersection between A = "the student is not tall" and B = "the student is not short" is
  
  [ A and B ] = "the student is not tall and not short" = "the student is medium in height"
  
  Height tall medium short
  
  Event A \(\checkmark\) \(\checkmark\)
  
  Event B \(\checkmark\) \(\checkmark\)
  
  Event [ A and B ] \(\times\) \(\checkmark\) \(\times\)

Height	tall	medium	short
Event A	\(\checkmark\)
Event B			\(\checkmark\)
Event [ A and B ]	\(\times\)	\(\times\)	\(\times\)

Height	tall	medium	short
Event A	\(\checkmark\)
Event B	\(\checkmark\)	\(\checkmark\)
Event [ A and B ]	\(\checkmark\)	\(\times\)	\(\times\)

Height	tall	medium	short
Event A		\(\checkmark\)	\(\checkmark\)
Event B	\(\checkmark\)	\(\checkmark\)
Event [ A and B ]	\(\times\)	\(\checkmark\)	\(\times\)

1.7.2 Example: Two Variables

If the experiment is about both Height and Weight, then
- the intersection between A = "the student is tall" and B = "the student is heavy" is
  
  [ A and B ] = "the student is tall and heavy"
  
  The Event heavy medium light
  
  tall [ A and B ] A A
  
  medium B
  
  short B
- the intersection between A = "the student is tall" and B = "the student is not heavy" is
  
  [ A and B ] = "the student is tall and not heavy"
  
  The Event heavy medium light
  
  tall A [ A and B ] [ A and B ]
  
  medium B B
  
  short B B
- the intersection between A = "the student is not tall" and B = "the student is not heavy" is
  
  [ A and B ] = "the student is not tall and not heavy"
  
  The Event heavy medium light
  
  tall B B
  
  medium A [ A and B ] [ A and B ]
  
  short A [ A and B ] [ A and B ]

The Event	heavy	medium	light
tall	[ A and B ]	A	A
medium	B
short	B

The Event	heavy	medium	light
tall	A	[ A and B ]	[ A and B ]
medium		B	B
short		B	B

The Event	heavy	medium	light
tall		B	B
medium	A	[ A and B ]	[ A and B ]
short	A	[ A and B ]	[ A and B ]

1.8 Union of Events

The union of events A and B is the event that occurs when either A or B occurs (ie, only A occurs, or only B occurs, or both A and B occur), which is denoted by [ A or B ].
The union event can be either a simple event or not, which depends on the situation.

1.8.1 Example: A Single Variable

If the experiment is about a single variable Height, then
- the union between A = "the student is tall" and B = "the student is short" is
  
  [ A or B ] = "the student is tall or short"
  
  Height tall medium short
  
  Event A \(\checkmark\)
  
  Event B \(\checkmark\)
  
  Event [ A or B ] \(\checkmark\) \(\times\) \(\checkmark\)
- the union between A = "the student is tall" and B = "the student is not short" is
  
  [ A or B ] = "the student is tall or not short"
  
  Height tall medium short
  
  Event A \(\checkmark\)
  
  Event B \(\checkmark\) \(\checkmark\)
  
  Event [ A or B ] \(\checkmark\) \(\checkmark\) \(\times\)
- the union between A = "the student is not tall" and B = "the student is not short" is
  
  [ A or B ] = "the student is not tall or not short"
  
  Height tall medium short
  
  Event A \(\checkmark\) \(\checkmark\)
  
  Event B \(\checkmark\) \(\checkmark\)
  
  Event [ A or B ] \(\checkmark\) \(\checkmark\) \(\checkmark\)

Height	tall	medium	short
Event A	\(\checkmark\)
Event B			\(\checkmark\)
Event [ A or B ]	\(\checkmark\)	\(\times\)	\(\checkmark\)

Height	tall	medium	short
Event A	\(\checkmark\)
Event B	\(\checkmark\)	\(\checkmark\)
Event [ A or B ]	\(\checkmark\)	\(\checkmark\)	\(\times\)

Height	tall	medium	short
Event A		\(\checkmark\)	\(\checkmark\)
Event B	\(\checkmark\)	\(\checkmark\)
Event [ A or B ]	\(\checkmark\)	\(\checkmark\)	\(\checkmark\)

1.8.2 Example: Two Variables

If the experiment is about both Height and Weight, then
- the union between A = "the student is tall" and B = "the student is heavy" is
  
  [ A or B ] = "the student is tall or heavy"
  
  The Event heavy medium light
  
  tall [ A or B ] [ A or B ] [ A or B ]
  
  medium [ A or B ]
  
  short [ A or B ]
- the union between A = "the student is tall" and B = "the student is not heavy" is
  
  [ A or B ] = "the student is tall or not heavy"
  
  The Event heavy medium light
  
  tall [ A or B ] [ A or B ] [ A or B ]
  
  medium [ A or B ] [ A or B ]
  
  short [ A or B ] [ A or B ]
- the union between A = "the student is not tall" and B = "the student is not heavy" is
  
  [ A or B ] = "the student is not tall or not heavy"
  
  The Event heavy medium light
  
  tall [ A or B ] [ A or B ]
  
  medium [ A or B ] [ A or B ] [ A or B ]
  
  short [ A or B ] [ A or B ] [ A or B ]

The Event	heavy	medium	light
tall	[ A or B ]	[ A or B ]	[ A or B ]
medium	[ A or B ]
short	[ A or B ]

The Event	heavy	medium	light
tall	[ A or B ]	[ A or B ]	[ A or B ]
medium		[ A or B ]	[ A or B ]
short		[ A or B ]	[ A or B ]

The Event	heavy	medium	light
tall		[ A or B ]	[ A or B ]
medium	[ A or B ]	[ A or B ]	[ A or B ]
short	[ A or B ]	[ A or B ]	[ A or B ]

1.9 Rule for Union, Intersection and Complement

[ A or B ]^C = [A^C and B^C].

That is, "either…or…" (ie. [ A or B ]) and "neither…nor…" (ie, [ A^C and B^C ]) are complementary to each other.
[ A and B ]^C = [A^C or B^C]

That is, "both…and…" (ie. [ A and B ]) and "at least one not" (ie, [ A^C or B^C ]) are complementary to each other.
The informal proofs of the above rules will be shown in the next section.

1.10 Rephrasing Complicated Events

Both A and B happen

\(\iff\) (A happens) and at the same time (B happens)

\(\iff\) [ A and B ] happens \(\boxed{\leftarrow\text{intersection}}\)

A A^C

B \(\checkmark\)

B^C
Either A or B happens

\(\iff\) (only A happens) or (only B happens) or (both A and B happens)

\(\iff\) [ A or B ] happens \(\boxed{\leftarrow\text{union}}\)

A A^C

B \(\checkmark\) \(\checkmark\)

B^C \(\checkmark\)
At least one of A and B happens

\(\iff\) (only A happens) or (only B happens) or (both A and B happens)

\(\iff\) [ A or B ] happens \(\boxed{\leftarrow\text{union}}\)

A A^C

B \(\checkmark\) \(\checkmark\)

B^C \(\checkmark\)
Neither A nor B happens

\(\iff\) (A does not happen) and at the same time (B does not happen)

\(\iff\) (A^C happens) and at the same time (B^C happens)

\(\iff\) both A^C and B^C happen

\(\iff\) [ A^C and B^C ] happens \(\boxed{\leftarrow\text{intersection of complements}}\)

A A^C

B

B^C \(\checkmark\)
Neither A nor B happens

\(\iff\) (A does not happen) and at the same time (B does not happen)

\(\iff\) the opposite of (at least one of A and B happens)

\(\iff\) the opposite of (either A or B happens)

\(\iff\) the opposite of (A or B happens)

\(\iff\) the opposite of [ A or B ] happens

\(\iff\) [ A or B ]^C happens \(\boxed{\leftarrow\text{complement of union}}\)

A A^C

B \(\times\) \(\times\)

B^C \(\times\) \(\checkmark\)
At most one of A and B happens

\(\iff\) (only A happens) or (only B happens) or (neither A nor B happens)

\(\iff\) opposite of (both A and B happen)

\(\iff\) (both A and B happen)^C

\(\iff\) [ A and B ]^C happens \(\boxed{\leftarrow\text{complement of intersection}}\)

A A^C

B \(\times\) \(\checkmark\)

B^C \(\checkmark\) \(\checkmark\)
At most one of A and B happens

\(\iff\) at least one of A and B does not happen

\(\iff\) (only A does not happen) or (only B does not happen) or (neither A nor B happens)

\(\iff\) (only A^C happens) or (only B^C happens) or (both A^C and B^C happen)

\(\iff\) [ A^C or B^C ] happens \(\boxed{\leftarrow\text{union of complements}}\)

A A^C

B \(\checkmark\)

B^C \(\checkmark\) \(\checkmark\)
Comment

From the "neither…nor…" cases and the "at most…" cases, we see that
- [ A or B ]^C = [A^C and B^C]
- [ A and B ]^C = [A^C or B^C]
which are just the two rules in the previous section.

2 References

Keller, Gerald. (2015). Statistics for Management and Economics, 10th Edition. Stamford: Cengage Learning.

	A	A^C
B	\(\checkmark\)
B^C

	A	A^C
B
B^C		\(\checkmark\)