Modeling Discrete Variables

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# Modeling Discrete Variables - PowerPoint PPT Presentation

Modeling Discrete Variables. Lecture 22-1 Sections 6.4 Wed, Mar 1, 2006. Two Types of Variable. Discrete variable – A variable whose set of possible values is a set of isolated points on the number line.

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### Modeling Discrete Variables

Lecture 22-1

Sections 6.4

Wed, Mar 1, 2006

Two Types of Variable
• Discrete variable – A variable whose set of possible values is a set of isolated points on the number line.
• Continuous variable – A variable whose set of possible values is a continuous interval of real numbers.
Example of a Discrete Variable
• Suppose that 10% of all households have no children, 30% have one child, 40% have two children, and 20% have three children.
• Select a household at random and let X = number of children.
• What is the distribution of X?
Example of a Discrete Variable
• We may list each value and its proportion.
• For 0.10 of the population, X = 0.
• For 0.30 of the population, X = 1.
• For 0.40 of the population, X = 2.
• For 0.20 of the population, X = 3.
Example of a Discrete Variable
• Or we may present it as a table.
Graphing a Discrete Variable
• Or we may present it as a stick graph.

P(X = x)

0.40

0.30

0.20

0.10

x

0

1

2

3

Graphing a Discrete Variable
• Or we may present it as a histogram.

P(X = x)

0.40

0.30

0.20

0.10

x

0

1

2

3

### Discrete Random Variables

Lecture 22-2

Section 7.5.1

Wed, Mar 1, 2006

Random Variables
• Random variable – A variable whose value is determined by the outcome of a procedure.
• The procedure includes at least one step whose outcome is left to chance.
• Therefore, the random variable takes on a new value each time the procedure is performed, even though the procedure is exactly the same.
Types of Random Variables
• Discrete Random Variable – A random variable whose set of possible values is a discrete set.
• Continuous Random Variable – A random variable whose set of possible values is a continuous set.
• The probability that something happens is the proportion of the time that it does happen out of all the times it was given an opportunity to happen.
• Therefore, “probability” and “proportion” are synonymous in the context of what we are doing.
Examples of Random Variables
• Roll two dice. Let X be the number of sixes.
• Possible values of X = {0, 1, 2}.
• Roll two dice. Let X be the total of the two numbers.
• Possible values of X = {2, 3, 4, …, 12}.
• Select a person at random and give him up to one hour to perform a simple task. Let X be the time it takes him to perform the task.
• Possible values of X are {x | 0 ≤ x ≤ 1}.
Discrete Probability Distribution Functions
• Discrete Probability Distribution Function (pdf) – A function that assigns a probability to each possible value of a discrete random variable.
Rolling Two Dice
• Roll two dice and let X be the number of sixes.
• Draw the 6  6 rectangle showing all 36 possibilities.
• From it we see that
• P(X = 0) = 25/36.
• P(X = 1) = 10/36.
• P(X = 2) = 1/36.
Rolling Two Dice
• We can summarize this in a table.
Example of a Discrete PDF
• Or we may present it as a stick graph.

P(X = x)

30/36

25/36

20/36

15/36

10/36

5/36

x

0

1

2

Example of a Discrete PDF
• Or we may present it as a histogram.

P(X = x)

30/36

25/36

20/36

15/36

10/36

5/36

x

0

1

2

Example of a Discrete PDF
• Suppose that 10% of all households have no children, 30% have one child, 40% have two children, and 20% have three children.
• Select a household at random and let X = number of children.
• Then X is a random variable.
• Which step in the procedure is left to chance?
• What is the pdf of X?
Example of a Discrete PDF
• We may present the pdf as a table.
Example of a Discrete PDF
• Or we may present it as a stick graph.

P(X = x)

0.40

0.30

0.20

0.10

x

0

1

2

3

Example of a Discrete PDF
• Or we may present it as a histogram.

P(X = x)

0.40

0.30

0.20

0.10

x

0

1

2

3