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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