February 16

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 = Thus, Pr(traf. fatality) = (about 1 in / )

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 ProbabilityVerbal expression 0.00Never 0.05Seldom 0.20Infrequent 0.50As often as not 0.80Very frequent 0.95Highly likely 1.00Always

§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: x01234 Pr(X=x)

Discrete Random Variables The pmf can be shown in tabular or graphical form x01234 Pr(X=x)

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 = = 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 – = “Four patients” pmf Ā A

Property 4 Illustrated Property 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) = = The probability of observing 3 or 4 successes is (about 73%). “Four patients” pmf B A

Mean and Variance of a Discrete RV Definitional formula for mean or expectation (p. 95) Definitional formula for variance (p. 96) Definitional formulas are advanced - not covered in some courses.

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 = “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”) Example: Pr(X ≤ 2) = shaded “tail” = =

§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) =

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.