Download 5: Probability Concepts

Chapter 5:
Probability Concepts
April 17
In Chapter 5:
5.1 What is Probability?
5.2 Types of Random Variables
5.3 Discrete Random Variables
5.4 Continuous Random Variables
5.5 More Rules and Properties of Probability
Definitions
• Random variable ≡ a numerical quantity that
takes on different values depending on chance
• Population ≡ the set of all possible values for a
random variable
• Event ≡ an outcome or set of outcomes
• Probability ≡ the proportion of times an event is
expected to occur in the population
Ideas about probability are founded on relative
frequencies (proportions) in populations.
Probability Illustrated
• In a given year, there were 42,636 traffic
fatalities in a population of N = 293,655,000
• If I randomly select a person from this
population, what is the probability they will
experience a traffic fatality by the end of that
year?
ANS: The relative frequency of this event in the
population = 42,636/ 293,655,000 = 0.0001452.
Thus, Pr(traf. fatality) = 0.0001452 (about 1 in
6887 1/.0001452)
Probability as a repetitive process
Experiments sample a population in which 20% of
observations are positives. This figure shows two such
experiments. The sample proportion approaches the true
probability of selection as n increases.
Subjective Probability
Probability can be used to quantify a level of belief
Probability
Verbal expression
0.00
Never
0.05
Seldom
0.20
Infrequent
0.50
As often as not
0.80
Very frequent
0.95
Highly likely
1.00
Always
§5.2: Random Variables
• Random variable ≡ a numerical quantity that
takes on different values depending on chance
• Two types of random variables
• Discrete random variables: a countable set of
possible outcome (e.g., the number of cases in
an SRS from the population)
• Continuous random variable: an unbroken
continuum of possible outcome (e.g., the
average weight of an SRS of newborns selected
from the population (Xeno’s paradox…)
•
§5.3: Discrete Random
Variables
Probability mass function (pmf) ≡ a
mathematical relation that assigns probabilities to
all possible outcomes for a discrete random
variables
• Illustrative example: “Four Patients”. Suppose I
treat four patients with an intervention that is
successful 75% of the time. Let X ≡ the variable
number of success in this experiment. This is the
pmf for this random variable:
x
0
1
2
3
4
Pr(X=x) 0.0039 0.0469 0.2109 0.4219 0.3164
Discrete Random Variables
The pmf can be shown in tabular or graphical form
x
0
1
2
3
4
Pr(X=x) 0.0039 0.0469 0.2109 0.4219 0.3164
Properties of Probabilities
• Property 1. Probabilities are always between 0
and 1
• Property 2. A sample space is all possible
outcomes. The probabilities in the sample space
sum to 1 (exactly).
• Property 3. The complement of an event is “the
event not happening”. The probability of a
complement is 1 minus the probability of the
event.
• Property 4. Probabilities of disjoint events can
be added.
Properties of Probabilities
In symbols
• Property 1. 0 ≤ Pr(A) ≤ 1
• Property 2. Pr(S) = 1, where S represent
the sample space (all possible outcomes)
• Property 3. Pr(Ā) = 1 – Pr(A), Ā represent
the complement of A (not A)
• Property 4. If A and B are disjoint, then
Pr(A or B) = Pr(A) + Pr(B)
Properties 1 & 2 Illustrated
“Four patients” pmf
Property 1. 0 ≤ Pr(A) ≤ 1
Note that all individual
probabilities are between 0
and 1.
Property 2. Pr(S) = 1
Note that the sum of all
probabilities = .0039 + .0469 +
.2109 + .4219 + .3164 = 1
Property 3 Illustrated
Property 3. Pr(Ā) = 1 – Pr(A),
“Four patients” pmf
As an example, let A
represent 4 successes.
Pr(A) = .3164
Ā
A
Let Ā represent the
complement of A (“not A”),
which is “3 or fewer”.
Pr(Ā) = 1 – Pr(A) = 1 – 0.3164
= 0.6836
Property 4 Illustrated
“Four patients” pmf
Property 4. Pr(A or B) = Pr(A)
+ Pr(B) for disjoint events
Let A represent 4 successes
Let B represent 3 successes
B A
Since A and B are disjoint,
Pr(A or B) = Pr(A) + Pr(B) =
0.3164 + 0.4129 = 0.7293.
The probability of observing 3
or 4 successes is 0.7293
(about 73%).
Area Under the Curve (AUC)
• The area under
curves (AUC) on a
pmf corresponds to
probability
• In this figure, Pr(X = 2)
= area of shaded
region = height ×
base = .2109 × 1.0 =
0.2109
“Four patients” pmf
Cumulative Probability
• “Cumulative
probability” refers to
probability of that value
or less
• Notation: Pr(X ≤ x)
• Corresponds to AUC to
the left of the point
(“left tail”)
.2109
.0469
.0039
Example: Pr(X ≤ 2) = shaded “tail”
= 0.0039 + 0.0469 + 0.2109 = 0.2617
§5.4 Continuous Random
Variables
Continuous random variables form a continuum of
possible values. As an illustration, consider the
spinner in this illustration. This spinner will generate
a continuum of random numbers between 0 to 1
§5.4: Continuous Random
Variables
A probability density functions (pdf) is a
mathematical relation that assigns probabilities to all
possible outcomes for a continuous random variable.
The pdf for our random spinner is shown here.
0.5
The shaded area
under the curve
represents probability,
in this instance:
Pr(0 ≤ X ≤ 0.5) = 0.5
Examples of pdfs
• pdfs obey all the rules of probabilities
• pdfs come in many forms (shapes). Here are
some examples:
Uniform pdf
Normal pdf
Chi-square pdf
Exercise 5.13 pdf
The most common pdf is the Normal. (We study the
Normal pdf in detail in the next chapter.)
Area Under the Curve
• As was the case with pmfs,
pdfs display probability with
the area under the curve
(AUC)
• This histogram shades bars
corresponding to ages ≤ 9
(~40% of histogram)
• This shaded AUC on the
Normal pdf curve also
corresponds to ~40% of total.
 x 

 
 12 
1
f ( x) 
e 
2 
2