chapter 5: probability concepts
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Chapter 5: Probability Concepts. 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. - PowerPoint PPT PresentationTRANSCRIPT
Apr 21, 2023
Chapter 5: Chapter 5: Probability ConceptsProbability Concepts
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 processExperiments 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
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
“Four patients” pmf
Property 3 Illustrated
Property 3. Pr(Ā) = 1 – Pr(A),
As an example, let A represent 4 successes.
Pr(A) = .3164
Let Ā represent the complement of A (“not A”), which is “3 or fewer”.
Pr(Ā) = 1 – Pr(A) = 1 – 0.3164 = 0.6836
“Four patients” pmf
Ā
A
Property 4 IllustratedProperty 4. Pr(A or B) = Pr(A) + Pr(B) for disjoint events
Let A represent 4 successes Let B represent 3 successes
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%).
“Four patients” pmf
B A
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”)
.0469
.2109
.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.
The shaded area under the curve represents probability, in this instance: Pr(0 ≤ X ≤ 0.5) = 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.
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