Download 1 Probability Models 2 Random Variables

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Probability Models
De…nition 1 A probability model is a mathematical description of an experiment consisting of two parts: a disjoint listing of all outcomes of the experiment
and an assignment of probabilities to each outcome.
A probability model is similar to a sample space but can describe more
complex outcomes because there is no longer the requirement that outcomes be
equally likely. Recall that the sample space for rolling a pair of dice has 36
outcomes.
(1; 1) (2; 1) (3; 1) (4; 1) (5; 1) (6; 1)
(1; 2) (2; 2) (3; 2) (4; 2) (5; 2) (6; 2)
(1; 3) (2; 3) (3; 3) (4; 3) (5; 3) (6; 3)
(1; 4) (2; 4) (3; 4) (4; 4) (5; 4) (6; 4)
(1; 5) (2; 5) (3; 5) (4; 5) (5; 5) (6; 5)
(1; 6) (2; 6) (3; 6) (4; 6) (5; 6) (6; 6)
Example 1 Construct the probability model for the sum of two dice.
sum x 2
3
4
5
6
7
1
2
3
4
5
6
P (x)
36
36
36
36
36
36
sum x 8
9
10 11 12
5
4
3
2
1
P (x)
36
36
36
36
36
Exercise 1 Construct the sample space for rolling one 3-sided die and one 4sided die. Construct the probability model for the sum of the two dice.
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Random Variables
De…nition 2 A random variable assumes a value based on the outcome of a
random event.
There are two types of random variables. Each type creates a very di¤erent
probability model with its own set of rules and computations.
De…nition 3 Discrete random variables can take one of a …nite number of
distinct outcomes. Discrete random variables jump from one state to the next
with nothing in between.
Example 2 The number of innings in a baseball game, number of players on
a basketball team and the number of cards in a deck are all discrete random
variables.
De…nition 4 Continuous random variables can take any numeric value
within a range of values.
Example 3 The weight of a baseball (in ounces), height of a player and the
time it takes (in seconds) to run 40 yards are all examples of continuous random
variables.
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Exercise 2 Consider an experiment whose population is the set of all Kennesaw
State University students. Which of these variables are discrete and which are
continuous?
number of classes the student is enrolled in this semester;
height in inches;
annual income in dollars;
GPA;
number of miles driven to campus.
Discrete probability models can use sample spaces to compute probabilities.
That does not work with continuous random variables which have an in…nite
number of values. The normal curve is an example of a continuous probability
model. In a continuous model, probability is computed as area under the
curve. As a general rule, such computations require calculus! Fortunately, the
normal curve is so important and well known there are many tools to compute
probabilities without resorting to calculus. In fact, that is what we were doing
every time we used Excel to perform computations in the normal curve.
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Exercises
1. Moore text: page 179: 9.26, 9.30, 9.34, 9.35, 9.38
2. For Kevin Garnett’s career stats (up to 2011), determine if each random
variable is continuous or discrete.
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games played;
games started;
minutes played;
…eld goals;
…eld goals attempted;
…eld goal percentage;
3-point …eld goals;
3-point …eld goals attempted;
3-point …eld goal percentage;
free throws;
free throw attempts;
free throw percentage;
o¤ensive rebounds;
defensive rebounds;
total rebounds;
assists;
steals;
blocks;
turnovers;
personal fouls;
blocks;
total points.
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