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STAT 2080
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Statistics

STAT 2080

John Walsh

Fall

Description

ST2080 Ali
4.1 Randomness
Consider outcomes from the following events:
- Tossing a coin toss
- Taking this course (STAT*2080)
In a single trial we cannot predict the outcome. However, there is a pattern that
emerges in the “long run”, i.e. over many repetitions.
A phenomenon is random if any individual outcome is uncertain, but there is a
regular distribution of outcomes from many repetitions.
Contrast the above events with the following:
- Tossing a coin 1,000 times and recording the top face on each toss
Ex. 4.3
Number Number Proportio
Tosser Tosses Heads n Heads
Count Buffon 4,040 2,048 0.5069
Karl Pearson 24,000 12,012 0.5005
John Kerrich 10,000 5,067 0.5067
- 1,000 students taking this course and recording each student’s grade
68 ST2080 Ali
We often use mathematical models to describe the distribution outcomes from
random events.
The probability of any outcome of a random event is the long run frequency
that the outcome would occur in a long series of repetitions.
To study random events we usually need outcomes from many
independent trials
The mathematical model is based on the empirical data, values
observed in real-life (through experiments or observational studies)
69 ST2080 Ali
4.2 Probability Models
The origin of developing mathematical models for random phenomenon comes
out of gambling and games of chance (roulette, poker, etc.). Formally,
A probability model is a mathematical description of a random phenomenon.
Probability models are comprised of the following:
A list of all possible outcomes, and
A probability associated with each outcome
We will first discuss how to describe the outcomes, and then look at some rules
of probability that will help in associating probabilities with outcomes.
Terminology
A sample space, S, is the set of all possible outcomes of a random
phenomenon.
Eg. Tossing a coin once and recording the top face.
S =
Eg. Tossing a six-sided die and recording the top face.
S =
Ex. 4.5 Close your eyes and let your pencil fall (tip down) on Table B.
Record the value of the number its tip lands on.
S =
70 ST2080 Ali
Eg. Record the top faces observed from tossing a coin three times.
S =
What if we instead record the number of heads obtained in three
tosses?
S =
An event is an outcome or set of outcomes of a random phenomenon, i.e. a
subset of the sample space.
Eg. Consider event that exactly two heads turn up in three tosses of a
coin. Call this event A. The set of outcomes associated with A are:
A =
Eg. Consider event that no more than two head turn up. Call this event B.
The set of outcomes associated with B are:
B =
Two disjoint events are two events that have no outcomes in common.
Eg. A: getting a 1 or 2 on a single toss of a die.
B: getting a 5 or 6 on a single toss of a die.
The complement of an event A is the set of all outcomes not in A.
Eg. A: getting an even number on a single toss of a die.
A :
71 ST2080 Ali
There are some basic rules of probability that we need in order to assign
probabilities to events.
All of these rules come out of the idea that probability is the long-run proportion
of repetitions on which an event occurs.
We will write each rule in words, and then in mathematical notation.
Rules of Probability
1. The probability of event A is always between 0 and 1.
i.e. P(A) satisfies 0 ≤ P(A) ≤ 1.
2. The sum of the probability of all possible outcomes is 1.
i.e. For sample space S: P(S) = 1.
3. If two events have no outcomes in common, the probability that one
or the other occurs is the sum of the individual probabilities.
i.e. For disjoint events A, B: P(A or B) = P(A) + P(B).
(Addition Rule)
4. The probability that an event does NOT occur equals 1 minus the
probability that the event does occur.
c
i.e. For event A: P(A ) = 1 – P(A).
(Complement Rule)
And the fifth rule is to come…
Let’s work through some examples first.
72 ST2080 Ali
Eg. A sociologist studies mobility in England by recording the social class
of a large sample of fathers and their sons. Social class is determined by
factors such as education and occupation, and is represented as a
variable with 5 levels: Class = {1 (lower class), 2, 3, 4, 5 (upper class)}.
Distribution of son’s social class for fathers in Class 1:
Son’s Class 1 2 3 4 5 Total
Probability 0.48 0.38 0.08 0.05 0.01
Consider the following events:
A = { son of Class 1 father is also in Class 1 }
B = {son of Class 1 father ends up in Class 4 or 5}
P(A) =
P(B) =
Note: for finite sample spaces, probability of an event equals the sum of
probabilities of all outcomes that make up that event.
A =
P(A ) =
Are A and B disjoint?
Q. What is the probability that the son of a father in the lower class either
remains in the lower class, or attains one of the top two classes?
73 ST2080 Ali
Ex. 4.12 Benford’s law. The first digit of numbers in legitimate expense
accounts or tax records tend to follow a distribution known as Benford’s
law. Investigators use this law to help detect fraudulent records.
First Digit: 1 2 3 4 5 6 7 8 9
Probability: 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
Consider events:
A = {first digit is 1}
B = {first digit is 6 or greater}
P(A) =
P(B) =
Note: event that first digit is “6 or greater” is different from “greater than 6”
P(first digit is anything other than 1) =
P(first digit is either equal to 1 OR 6 or greater )
=
74 ST2080 Ali
Ex. 4.14 Benford’s law (cont’d). Consider the event C:
C = {first digit is odd}
P(C) =
P(B or C) =
S
If the probability of each outcome is equal, then the probability of each

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