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Statistical SciencesCourse Code

STA257H1Professor

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RANDOM VARIABLES AND EVENTS

Definition: Random Variable

A random variable is a variable that is random. They are denoted X, Y, Z.

Note

A random variable which is constant will be called a constant random variable.

Example

Roll a “fair” die and let X denote the number of dots.

• X is a random variable. Its possible values are 1, 2, 3, 4, 5, 6.

• Observing X leads to data or observed value. We will denote a typical observation by x.

Definition: Event

In general, an event is a statement involving random variables. They are denoted A, B, C, D.

Notation

{X is even} means “the event that X is even”.

Note

Events either occur or they don’t.

Notes

1) Events may be thought of as collections/sets of outcomes.

2) The sure event (denoted S or Ω) is the set of all possible outcomes.

3) S includes all the outcomes so that any event A is made up of outcomes “drawn” from S. A is a subset of S and

we write SA

⊂

.

4) The impossible event or empty set (denoted

φ

) is the event which consists of no possible outcomes and hence

never occurs.

• Notice: SA

⊂

⊂

φ

.

5) We can talk about any function of X, say,

(

)

Xg. Examples: 2

X

, Xsin,

(

)

XeXexp=, etc.

Definition: The Indicator Random Variable or Bernoulli Random Variable

If A is an event, then we define the indicator random variable or Bernoulli random variable for/of A to be

=otherwise 0

occurs if 1A

IA.

Example

Let X be a random variable. Measure it an infinite number of times to get ,,, 321 xxx .

• The average will be the expected value or expectation of X, denoted

(

)

XE .

• If X denotes marks, then

{

}

{

}

0pass ≥== XA .

• The indicator random variable might look like: ,0,1,0,0.

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Definition

The proportion or the relative frequency of 1s will be the probability of A, denoted

(

)

AP .

• Notice

(

)

(

)

A

IEAP =.

MORE ON EVENTS AND RANDOM VARIABLES

• Events consist of outcomes.

• Every event is a subset of S.

Sometimes the same event has different descriptions, so we might want to show something like:

BA

AB

BA =

⊂

⊂.

How to show

B

A

⊂

?

• Take an arbitrary element of A.

• Show that it is an element of B, i.e. let Ax

∈

and show Bx

∈

.

From our point of view,

B

A

⊂

means BA

.

Two events are equal (i.e. the same) if BA

and AB

.

Random Variables

How to show two random variables X and Y are equal? They are equal if they are equal all the time.

Example

Let A1 and A2 be disjoint events (i.e. no outcomes in common, no overlap, etc.). Look at the event 21 AA ∪

(the event that at least one of the A’s occurs).

• The indicator random variable of this new event is 21 AA

I∪ or

(

)

21 AAI∪. “Clearly”,

2121 AAAA III +=

∪.

Notes

1)

∪∪=

∞

=

21

1

AAA

i

i is the event that at least one of the A’s occurs.

2) If the A’s are disjoint, then a common notation for

++=

∞

=

21

1

AAA

i

i.

3)

2121

1

AAAAA

i

i=∩∩=

∞

=

is the event that all of the A’s occur.

COMPLEMENTS

Definition

The complement of an event A is the event consisting of all outcomes not in A. It is denoted c

A

or

A

.

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Example

Toss a coin n times and set heads ofnumber =Y.

•

{

}

{

}

1head oneleast at ≥== YA.

•

{

}

{

}

{

}

1 tailsallhead no ==== YAc.

Set

{

}

first toss on the head

1=A,

{

}

tosssecond on the head

2=A, etc.

•

{

}

mYAAA m==

21 .

•

{

}

1

21 ≥=∪∪∪ YAAA m

.

•

(

)

{

}

c

m

ccc

mAAAYAAA 2121 0===∪∪∪ .

In General

•

i

c

i

c

i

iAA =

and

i

c

i

c

i

iAA =

– Morgan’s Laws.

•

(

)

AA c

c=.

In terms of indicator random variables,

• 1=+ A

AII c.

• m

i

i

AA

A

III

1

1

=

∞

=

.

• 212121 AAAAAA IIII ∩∪ −+= .

BERNOULLI (P) RANDOM VARIABLES

Definition

X is a Bernoulli (p) random variable if X can only take on either 0 or 1 and

(

)

pXP ==1.

Note

1) Let X be Bernoulli (p) (X ~ Bernoulli (p)). Now measure it an infinite number of times. You end up with a list

of 0’s and 1’s. The proportion of 1’s will be p and the proportion of 0’s will be q.

2) Now,

•

(

)

pXE = (1st moment of X),

(

)

pXE =

2 (2nd moment of X),

(

)

pXE n=.

•

(

)

(

)

(

)

122 qpEX+= ,

(

)

(

)

(

)

133 qpEX+= .

3) If s is a dummy variable, then

(

)

qpssEX+= . Call

(

)

X

sE the probability generation function denoted by

(

)

sG. This function is in fact defined for any count random variable.

KOLMOGOROV AXIOMS/LAWS OF PROBABILTY

1)

(

)

0≥AP ;

(

)

1=SP.

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